Discussion:
Packaging all the axioms of arithmetic and algebra into a matrix columns
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Archimedes Plutonium
2017-08-11 13:13:01 UTC
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In these columns of 2byN matrices, I propose that they alone are all the axioms that embody Arithmetic and Algebra. Here forth, we dispense of the Peano axioms for Counting Numbers and simply cite these Matrices as the axioms of both Arithmetic and Algebra. Now, I propose for Geometry, the axioms are all embodied by a rectangle with its two diagonals and the triangles of right triangles with Pythagorean Theorem. What is true of those figures is all the axioms of Geometry, two points determine a line, opposite sides parallel due to 90 degree perpendicular. Now, the real challenging Schemata for all the axioms of a subject, is logic, where, if we can do a Arithmetic schemata of all the axioms of Arithmetic, we should be able to do a schemata of all the Logic axioms, the if then, or, and, equal, etc.

So, these schemata replace the axioms of Arithmetic and Algebra, replace the axioms of Geometry by a rectangle schemata, and replace the axioms of Logic, by a truth table schemata of the 4 possibilities::

Logic Schemata of all its axioms
TTTT, TFFF, TFFT, TFUU

Arithmetic and Algebra schemata of all its axioms


0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6

Now, in another post, I show how the Matrix Column Schemata delivers Mathematical Induction axiom, and easily delivers Successor Axiom. What the Old Math Peano axioms fail in providing is uniqueness of sum or product, fail in Complementarity , and fail in Symmetry.

Now, to package all the Geometry axioms, package them into a Schemata, we need just a figure of the Rectangle with its diagonals, for that provides us with the usual axioms of two points determine a line, the parallel postulate and most important the Pythagorean theorem. But it provides something special that Euclid and Hilbert axioms missed, for they do not axiomatize angles, provide for angles as axioms in their geometry, nor do their axioms provide for a Coordinate System, which a Rectangle as a Square provides for a axiom of Coordinate System.

___________
| |
| |
|__________|

when diagonals are included and angle notation, we thus provide all of Geometry Axioms that exist with a Schemata. So in the proof of a geometry theorem, when we come to a justification such as parallel line axiom, we can cite that, or we can cite rectangle-schemata.

Now, some will wonder how do they represent Hyperbolic geometry or Elliptic geometry with a Axiom Schemata? And I would answer them, that there never was three independent geometries where the parallel postulate is different, with greater than, less than or equal to 180 degrees triangles. The triangle on the sphere globe of 90 + 90 +90. There is no separate independent three geometries. All geometries are just plain Euclidean geometry. For, when you take a block and carve out a sphere representing Elliptic geometry and the hole in the block where the sphere is carved out of is Hyperbolic geometry, we realize that Elliptic and Hyperbolic were cut-outs, always cut-outs of Euclidean geometry.

The formula is true as ever-- Euclidean Geometry = Elliptic unioned with Hyperbolic geometry.

Old Math was mesmerized with the Lobachevskian and Bolyai NonEuclidean hyperbolic geometries of circa 1830, and Riemann with elliptic geometry circa 1854.

These are not independent separate geometries and do not deserve separate axioms, for they are complementaries of one another that are cut-outs of just plain Euclidean Geometry.

So, in our Rectangle Schemata, should we draw a diagonal curved arc, then the two triangles formed, one is hyperbolic and the other is elliptic, but both together are just Euclidean.

So, here, what I am doing is distilling a Schemata, that embodies all the axioms of a given Subject matter, be it Arithmetic, Algebra, Geometry, and even Logic. A schemata that one just refers to the schemata for the axiom that covers a line of a proof justification.

And this is beautiful way of announcing the axioms of Counting Numbers, for we thus know symmetry is embodied, that uniqueness is embodied, that Complementarity is embodied. And we can spell out specific traits like commutative or mathematical induction, or, successor.

AP
q***@gmail.com
2017-08-11 15:56:55 UTC
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Post by Archimedes Plutonium
In these columns of 2byN matrices, I propose that they alone are all the axioms that embody Arithmetic and Algebra. Here forth, we dispense of the Peano axioms for Counting Numbers and simply cite these Matrices as the axioms of both Arithmetic and Algebra. Now, I propose for Geometry, the axioms are all embodied by a rectangle with its two diagonals and the triangles of right triangles with Pythagorean Theorem. What is true of those figures is all the axioms of Geometry, two points determine a line, opposite sides parallel due to 90 degree perpendicular. Now, the real challenging Schemata for all the axioms of a subject, is logic, where, if we can do a Arithmetic schemata of all the axioms of Arithmetic, we should be able to do a schemata of all the Logic axioms, the if then, or, and, equal, etc.
Now your delusions of grandiosity are replacing all of
arithmetic, algebra, geometry and logic with your tables of calculations.

But only a day or two ago, when calculations again and again and again
showed that your "intersections of cones and planes is never an ellipse"
was false and demolished your claim that you have screeched for years,
you tried to dismiss all this with "calculations, but no logic, where
is the logic?"

Yet yesterday you were begging for someone to do more calculations for
you because you are thinking a few more calculations would somehow be
your proof of Goldbach for all even N out to your infinity and beyond.

A few calculations can often demolish one of your nonsense claims,
but calculations are almost never going to prove any of your nonsense claims.

Never ever ever bring a crank to a brain fight
Post by Archimedes Plutonium
Logic Schemata of all its axioms
TTTT, TFFF, TFFT, TFUU
Arithmetic and Algebra schemata of all its axioms
0  1
0  2
1  1
0  3
1  2
0  4
1  3
2  2
0  5
1  4
2  3
0  6
1  5
2  4
3  3
0  7
1  6
2  5
3  4
0  8
1  7
2  6
3  5
4  4
0  9
1  8
2  7
3  6
4  5
0  10
1   9
2   8
3   7
4   6
5   5
0   11
1   10
2    9
3    8
4    7
5    6
0    12
1    11
2    10
3      9
4      8
5      7
6      6
Now, in another post, I show how the Matrix Column Schemata delivers Mathematical Induction axiom, and easily delivers Successor Axiom. What the Old Math Peano axioms fail in providing is uniqueness of sum or product, fail in Complementarity , and fail in Symmetry.
Now, to package all the Geometry axioms, package them into a Schemata, we need just a figure of the Rectangle with its diagonals, for that provides us with the usual axioms of two points determine a line, the parallel postulate and most important the Pythagorean theorem. But it provides something special that Euclid and Hilbert axioms missed, for they do not axiomatize angles, provide for angles as axioms in their geometry, nor do their axioms provide for a Coordinate System, which a Rectangle as a Square provides for a axiom of Coordinate System.
___________
| |
| |
|__________|
when diagonals are included and angle notation, we thus provide all of Geometry Axioms that exist with a Schemata. So in the proof of a geometry theorem, when we come to a justification such as parallel line axiom, we can cite that, or we can cite rectangle-schemata.
Now, some will wonder how do they represent Hyperbolic geometry or Elliptic geometry with a Axiom Schemata? And I would answer them, that there never was three independent geometries where the parallel postulate is different, with greater than, less than or equal to 180 degrees triangles. The triangle on the sphere globe of 90 + 90 +90. There is no separate independent three geometries. All geometries are just plain Euclidean geometry. For, when you take a block and carve out a sphere representing Elliptic geometry and the hole in the block where the sphere is carved out of is Hyperbolic geometry, we realize that Elliptic and Hyperbolic were cut-outs, always cut-outs of Euclidean geometry.
The formula is true as ever-- Euclidean Geometry = Elliptic unioned with Hyperbolic geometry.
Old Math was mesmerized with the Lobachevskian and Bolyai NonEuclidean hyperbolic geometries of circa 1830, and Riemann with elliptic geometry circa 1854.
These are not independent separate geometries and do not deserve separate axioms, for they are complementaries of one another that are cut-outs of just plain Euclidean Geometry.
So, in our Rectangle Schemata, should we draw a diagonal curved arc, then the two triangles formed, one is hyperbolic and the other is elliptic, but both together are just Euclidean.
So, here, what I am doing is distilling a Schemata, that embodies all the axioms of a given Subject matter, be it Arithmetic, Algebra, Geometry, and even Logic. A schemata that one just refers to the schemata for the axiom that covers a line of a proof justification.
And this is beautiful way of announcing the axioms of Counting Numbers, for we thus know symmetry is embodied, that uniqueness is embodied, that Complementarity is embodied. And we can spell out specific traits like commutative or mathematical induction, or, successor.
AP
Vinicius Claudino Ferraz
2017-08-11 18:40:46 UTC
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Hey, AP, may I already delete claudino.webs.com/temp.txt ?

Now there's a list where "2n - 3 is prime" or not.
What do you think?

6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
12
14 = 3 + 11
16 = 3 + 13
18
20 = 3 + 17
22 = 3 + 19
24
26 = 3 + 23
28
30
32 = 3 + 29
34 = 3 + 31
36
38
40 = 3 + 37
42
44 = 3 + 41
46 = 3 + 43
48
50 = 3 + 47
52
54
56 = 3 + 53
58
60
62 = 3 + 59
64 = 3 + 61
66
68
70 = 3 + 67
72
74 = 3 + 71
76 = 3 + 73
78
80
82 = 3 + 79
84
86 = 3 + 83
88
90
92 = 3 + 89
94
96
98
100 = 3 + 97
102
104 = 3 + 101
106 = 3 + 103
108
110 = 3 + 107
112 = 3 + 109
114
116 = 3 + 113
118
120
122
124
126
128
130 = 3 + 127
132
134 = 3 + 131
136
138
140 = 3 + 137
142 = 3 + 139
144
146
148
150
152 = 3 + 149
154 = 3 + 151
156
158
160 = 3 + 157
162
164
166 = 3 + 163
168
170 = 3 + 167
172
174
176 = 3 + 173
178
180
182 = 3 + 179
184 = 3 + 181
186
188
190
192
194 = 3 + 191
196 = 3 + 193
198
200 = 3 + 197
202 = 3 + 199
204
206
208
210
212
214 = 3 + 211
216
218
220
222
224
226 = 3 + 223
228
230 = 3 + 227
232 = 3 + 229
234
236 = 3 + 233
238
240
242 = 3 + 239
244 = 3 + 241
246
248
250
252
254 = 3 + 251
256
258
260 = 3 + 257
262
264
266 = 3 + 263
268
270
272 = 3 + 269
274 = 3 + 271
276
278
280 = 3 + 277
282
284 = 3 + 281
286 = 3 + 283
288
290
292
294
296 = 3 + 293
298
300
302
304
306
308
310 = 3 + 307
312
314 = 3 + 311
316 = 3 + 313
318
320 = 3 + 317
322
324
326
328
330
332
334 = 3 + 331
336
338
340 = 3 + 337
342
344
346
348
350 = 3 + 347
352 = 3 + 349
354
356 = 3 + 353
358
360
362 = 3 + 359
364
366
368
370 = 3 + 367
372
374
376 = 3 + 373
378
380
382 = 3 + 379
384
386 = 3 + 383
388
390
392 = 3 + 389
394
396
398
400 = 3 + 397
402
404 = 3 + 401
406
408
410
412 = 3 + 409
414
416
418
420
422 = 3 + 419
424 = 3 + 421
426
428
430
432
434 = 3 + 431
436 = 3 + 433
438
440
442 = 3 + 439
444
446 = 3 + 443
448
450
452 = 3 + 449
454
456
458
460 = 3 + 457
462
464 = 3 + 461
466 = 3 + 463
468
470 = 3 + 467
472
474
476
478
480
482 = 3 + 479
484
486
488
490 = 3 + 487
492
494 = 3 + 491
496
498
500
502 = 3 + 499
504
506 = 3 + 503
508
510
512 = 3 + 509
514
516
518
520
522
524 = 3 + 521
526 = 3 + 523
528
530
532
534
536
538
540
542
544 = 3 + 541
546
548
550 = 3 + 547
552
554
556
558
560 = 3 + 557
562
564
566 = 3 + 563
568
570
572 = 3 + 569
574 = 3 + 571
576
578
580 = 3 + 577
582
584
586
588
590 = 3 + 587
592
594
596 = 3 + 593
598
600
602 = 3 + 599
604 = 3 + 601
606
608
610 = 3 + 607
612
614
616 = 3 + 613
618
620 = 3 + 617
622 = 3 + 619
624
626
628
630
632
634 = 3 + 631
636
638
640
642
644 = 3 + 641
646 = 3 + 643
648
650 = 3 + 647
652
654
656 = 3 + 653
658
660
662 = 3 + 659
664 = 3 + 661
666
668
670
672
674
676 = 3 + 673
678
680 = 3 + 677
682
684
686 = 3 + 683
688
690
692
694 = 3 + 691
696
698
700
702
704 = 3 + 701
706
708
710
712 = 3 + 709
714
716
718
720
722 = 3 + 719
724
726
728
730 = 3 + 727
732
734
736 = 3 + 733
738
740
742 = 3 + 739
744
746 = 3 + 743
748
750
752
754 = 3 + 751
756
758
760 = 3 + 757
762
764 = 3 + 761
766
768
770
772 = 3 + 769
774
776 = 3 + 773
778
780
782
784
786
788
790 = 3 + 787
792
794
796
798
800 = 3 + 797
802
804
806
808
810
812 = 3 + 809
814 = 3 + 811
816
818
820
822
824 = 3 + 821
826 = 3 + 823
828
830 = 3 + 827
832 = 3 + 829
834
836
838
840
842 = 3 + 839
844
846
848
850
852
854
856 = 3 + 853
858
860 = 3 + 857
862 = 3 + 859
864
866 = 3 + 863
868
870
872
874
876
878
880 = 3 + 877
882
884 = 3 + 881
886 = 3 + 883
888
890 = 3 + 887
892
894
896
898
900
902
904
906
908
910 = 3 + 907
912
914 = 3 + 911
916
918
920
922 = 3 + 919
924
926
928
930
932 = 3 + 929
934
936
938
940 = 3 + 937
942
944 = 3 + 941
946
948
950 = 3 + 947
952
954
956 = 3 + 953
958
960
962
964
966
968
970 = 3 + 967
972
974 = 3 + 971
976
978
980 = 3 + 977
982
984
986 = 3 + 983
988
990
992
994 = 3 + 991
996
998
1000 = 3 + 997
1002
1004
1006
1008
1010
1012 = 3 + 1009
1014
1016 = 3 + 1013
1018
1020
1022 = 3 + 1019
1024 = 3 + 1021
1026
1028
1030
1032
1034 = 3 + 1031
1036 = 3 + 1033
1038
1040
1042 = 3 + 1039
1044
1046
1048
1050
1052 = 3 + 1049
1054 = 3 + 1051
1056
1058
1060
1062
1064 = 3 + 1061
1066 = 3 + 1063
1068
1070
1072 = 3 + 1069
1074
1076
1078
1080
1082
1084
1086
1088
1090 = 3 + 1087
1092
1094 = 3 + 1091
1096 = 3 + 1093
1098
1100 = 3 + 1097
1102
1104
1106 = 3 + 1103
1108
1110
1112 = 3 + 1109
1114
1116
1118
1120 = 3 + 1117
1122
1124
1126 = 3 + 1123
1128
1130
1132 = 3 + 1129
1134
1136
1138
1140
1142
1144
1146
1148
1150
1152
1154 = 3 + 1151
1156 = 3 + 1153
1158
1160
1162
1164
1166 = 3 + 1163
1168
1170
1172
1174 = 3 + 1171
1176
1178
1180
1182
1184 = 3 + 1181
1186
1188
1190 = 3 + 1187
1192
1194
1196 = 3 + 1193
1198
1200
1202
1204 = 3 + 1201
1206
1208
1210
1212
1214
1216 = 3 + 1213
1218
1220 = 3 + 1217
1222
1224
1226 = 3 + 1223
1228
1230
1232 = 3 + 1229
1234 = 3 + 1231
1236
1238
1240 = 3 + 1237
1242
1244
1246
1248
1250
1252 = 3 + 1249
1254
1256
1258
1260
1262 = 3 + 1259
1264
1266
1268
1270
1272
1274
1276
1278
1280 = 3 + 1277
1282 = 3 + 1279
1284
1286 = 3 + 1283
1288
1290
1292 = 3 + 1289
1294 = 3 + 1291
1296
1298
1300 = 3 + 1297
1302
1304 = 3 + 1301
1306 = 3 + 1303
1308
1310 = 3 + 1307
1312
1314
1316
1318
1320
1322 = 3 + 1319
1324 = 3 + 1321
1326
1328
1330 = 3 + 1327
1332
1334
1336
1338
1340
1342
1344
1346
1348
1350
1352
1354
1356
1358
1360
1362
1364 = 3 + 1361
1366
1368
1370 = 3 + 1367
1372
1374
1376 = 3 + 1373
1378
1380
1382
1384 = 3 + 1381
1386
1388
1390
1392
1394
1396
1398
1400
1402 = 3 + 1399
1404
1406
1408
1410
1412 = 3 + 1409
1414
1416
1418
1420
1422
1424
1426 = 3 + 1423
1428
1430 = 3 + 1427
1432 = 3 + 1429
1434
1436 = 3 + 1433
1438
1440
1442 = 3 + 1439
1444
1446
1448
1450 = 3 + 1447
1452
1454 = 3 + 1451
1456 = 3 + 1453
1458
1460
1462 = 3 + 1459
1464
1466
1468
1470
1472
1474 = 3 + 1471
1476
1478
1480
1482
1484 = 3 + 1481
1486 = 3 + 1483
1488
1490 = 3 + 1487
1492 = 3 + 1489
1494
1496 = 3 + 1493
1498
1500
1502 = 3 + 1499
1504
1506
1508
1510
1512
1514 = 3 + 1511
1516
1518
1520
1522
1524
1526 = 3 + 1523
1528
1530
1532
1534 = 3 + 1531
1536
1538
1540
1542
1544
1546 = 3 + 1543
1548
1550
1552 = 3 + 1549
1554
1556 = 3 + 1553
1558
1560
1562 = 3 + 1559
1564
1566
1568
1570 = 3 + 1567
1572
1574 = 3 + 1571
1576
1578
1580
1582 = 3 + 1579
1584
1586 = 3 + 1583
1588
1590
1592
1594
1596
1598
1600 = 3 + 1597
1602
1604 = 3 + 1601
1606
1608
1610 = 3 + 1607
1612 = 3 + 1609
1614
1616 = 3 + 1613
1618
1620
1622 = 3 + 1619
1624 = 3 + 1621
1626
1628
1630 = 3 + 1627
1632
1634
1636
1638
1640 = 3 + 1637
1642
1644
1646
1648
1650
1652
1654
1656
1658
1660 = 3 + 1657
1662
1664
1666 = 3 + 1663
1668
1670 = 3 + 1667
1672 = 3 + 1669
1674
1676
1678
1680
1682
1684
1686
1688
1690
1692
1694
1696 = 3 + 1693
1698
1700 = 3 + 1697
1702 = 3 + 1699
1704
1706
1708
1710
1712 = 3 + 1709
1714
1716
1718
1720
1722
1724 = 3 + 1721
1726 = 3 + 1723
1728
1730
1732
1734
1736 = 3 + 1733
1738
1740
1742
1744 = 3 + 1741
1746
1748
1750 = 3 + 1747
1752
1754
1756 = 3 + 1753
1758
1760
1762 = 3 + 1759
1764
1766
1768
1770
1772
1774
1776
1778
1780 = 3 + 1777
1782
1784
1786 = 3 + 1783
1788
1790 = 3 + 1787
1792 = 3 + 1789
1794
1796
1798
1800
1802
1804 = 3 + 1801
1806
1808
1810
1812
1814 = 3 + 1811
1816
1818
1820
1822
1824
1826 = 3 + 1823
1828
1830
1832
1834 = 3 + 1831
1836
1838
1840
1842
1844
1846
1848
1850 = 3 + 1847
1852
1854
1856
1858
1860
1862
1864 = 3 + 1861
1866
1868
1870 = 3 + 1867
1872
1874 = 3 + 1871
1876 = 3 + 1873
1878
1880 = 3 + 1877
1882 = 3 + 1879
1884
1886
1888
1890
1892 = 3 + 1889
1894
1896
1898
1900
1902
1904 = 3 + 1901
1906
1908
1910 = 3 + 1907
1912
1914
1916 = 3 + 1913
1918
1920
1922
1924
1926
1928
1930
1932
1934 = 3 + 1931
1936 = 3 + 1933
1938
1940
1942
1944
1946
1948
1950
1952 = 3 + 1949
1954 = 3 + 1951
1956
1958
1960
1962
1964
1966
1968
1970
1972
1974
1976 = 3 + 1973
1978
1980
1982 = 3 + 1979
1984
1986
1988
1990 = 3 + 1987
1992
1994
1996 = 3 + 1993
1998
2000 = 3 + 1997
2002 = 3 + 1999
2004
2006 = 3 + 2003
2008
2010
2012
2014 = 3 + 2011
2016
2018
2020 = 3 + 2017
2022
2024
2026
2028
2030 = 3 + 2027
2032 = 3 + 2029
2034
2036
2038
2040
2042 = 3 + 2039
2044
2046
2048
2050
2052
2054
2056 = 3 + 2053
2058
2060
2062
2064
2066 = 3 + 2063
2068
2070
2072 = 3 + 2069
2074
2076
2078
2080
2082
2084 = 3 + 2081
2086 = 3 + 2083
2088
2090 = 3 + 2087
2092 = 3 + 2089
2094
2096
2098
2100
2102 = 3 + 2099
2104
2106
2108
2110
2112
2114 = 3 + 2111
2116 = 3 + 2113
2118
2120
2122
2124
2126
2128
2130
2132 = 3 + 2129
2134 = 3 + 2131
2136
2138
2140 = 3 + 2137
2142
2144 = 3 + 2141
2146 = 3 + 2143
2148
2150
2152
2154
2156 = 3 + 2153
2158
2160
2162
2164 = 3 + 2161
2166
2168
2170
2172
2174
2176
2178
2180
2182 = 3 + 2179



Em sexta-feira, 11 de agosto de 2017 10:13:15 UTC-3, Archimedes Plutonium escreveu:
Archimedes Plutonium
2017-08-11 21:15:26 UTC
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Post by Vinicius Claudino Ferraz
Hey, AP, may I already delete claudino.webs.com/temp.txt ?
Yes, please delete that, for I do not visit websites.

Please go beyond this 95 in this table and post it to sci.math


3*3 = 9                 3+3= 6
3 * 5  = 15           3+5 = 8
3 * 7  = 21           3+7 = 10
5 * 5  = 25           5+5 =10
3 * 11 = 33          3+11 = 14
5 * 7  = 35           5+ 7 = 12
3 * 13 = 39          3+13 = 16
7 *  7  = 49          7+7 = 14
3 * 17 = 51          3+17 = 20
5 * 11 = 55          5+11 = 16
3 * 19 = 57          3+19 = 22
5 * 13  = 65         5+13 = 18
3 * 23  = 69         3+23 = 26
7 * 11  = 77         7+11 = 18
5 *17  = 85          5+17 = 22
3 * 29  = 87        3+29 = 32
7 * 13  =  91       7+13 = 20
3 * 31  =  93        3+ 31 = 34
5 * 19  =  95        5+ 19 = 24

a
Vinicius Claudino Ferraz
2017-08-12 15:13:56 UTC
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3 * 5 = 15 3 + 5 = 8
3 * 7 = 21 3 + 7 = 10
3 * 11 = 33 3 + 11 = 14
3 * 13 = 39 3 + 13 = 16
3 * 17 = 51 3 + 17 = 20
3 * 19 = 57 3 + 19 = 22
3 * 23 = 69 3 + 23 = 26
3 * 29 = 87 3 + 29 = 32
3 * 31 = 93 3 + 31 = 34
3 * 37 = 111 3 + 37 = 40
3 * 41 = 123 3 + 41 = 44
3 * 43 = 129 3 + 43 = 46
3 * 47 = 141 3 + 47 = 50
3 * 53 = 159 3 + 53 = 56
3 * 59 = 177 3 + 59 = 62
3 * 61 = 183 3 + 61 = 64
3 * 67 = 201 3 + 67 = 70
3 * 71 = 213 3 + 71 = 74
3 * 73 = 219 3 + 73 = 76
3 * 79 = 237 3 + 79 = 82
3 * 83 = 249 3 + 83 = 86
3 * 89 = 267 3 + 89 = 92
3 * 97 = 291 3 + 97 = 100
3 * 101 = 303 3 + 101 = 104
3 * 103 = 309 3 + 103 = 106
3 * 107 = 321 3 + 107 = 110
3 * 109 = 327 3 + 109 = 112
3 * 113 = 339 3 + 113 = 116
3 * 127 = 381 3 + 127 = 130
3 * 131 = 393 3 + 131 = 134
3 * 137 = 411 3 + 137 = 140
3 * 139 = 417 3 + 139 = 142
3 * 149 = 447 3 + 149 = 152
3 * 151 = 453 3 + 151 = 154
3 * 157 = 471 3 + 157 = 160
3 * 163 = 489 3 + 163 = 166
3 * 167 = 501 3 + 167 = 170
3 * 173 = 519 3 + 173 = 176
3 * 179 = 537 3 + 179 = 182
3 * 181 = 543 3 + 181 = 184
3 * 191 = 573 3 + 191 = 194
3 * 193 = 579 3 + 193 = 196
3 * 197 = 591 3 + 197 = 200
3 * 199 = 597 3 + 199 = 202
3 * 211 = 633 3 + 211 = 214
3 * 223 = 669 3 + 223 = 226
3 * 227 = 681 3 + 227 = 230
3 * 229 = 687 3 + 229 = 232
3 * 233 = 699 3 + 233 = 236
3 * 239 = 717 3 + 239 = 242
5 * 7 = 35 5 + 7 = 12
5 * 11 = 55 5 + 11 = 16
5 * 13 = 65 5 + 13 = 18
5 * 17 = 85 5 + 17 = 22
5 * 19 = 95 5 + 19 = 24
5 * 23 = 115 5 + 23 = 28
5 * 29 = 145 5 + 29 = 34
5 * 31 = 155 5 + 31 = 36
5 * 37 = 185 5 + 37 = 42
5 * 41 = 205 5 + 41 = 46
5 * 43 = 215 5 + 43 = 48
5 * 47 = 235 5 + 47 = 52
5 * 53 = 265 5 + 53 = 58
5 * 59 = 295 5 + 59 = 64
5 * 61 = 305 5 + 61 = 66
5 * 67 = 335 5 + 67 = 72
5 * 71 = 355 5 + 71 = 76
5 * 73 = 365 5 + 73 = 78
5 * 79 = 395 5 + 79 = 84
5 * 83 = 415 5 + 83 = 88
5 * 89 = 445 5 + 89 = 94
5 * 97 = 485 5 + 97 = 102
5 * 101 = 505 5 + 101 = 106
5 * 103 = 515 5 + 103 = 108
5 * 107 = 535 5 + 107 = 112
5 * 109 = 545 5 + 109 = 114
5 * 113 = 565 5 + 113 = 118
5 * 127 = 635 5 + 127 = 132
5 * 131 = 655 5 + 131 = 136
5 * 137 = 685 5 + 137 = 142
5 * 139 = 695 5 + 139 = 144
5 * 149 = 745 5 + 149 = 154
5 * 151 = 755 5 + 151 = 156
5 * 157 = 785 5 + 157 = 162
5 * 163 = 815 5 + 163 = 168
5 * 167 = 835 5 + 167 = 172
5 * 173 = 865 5 + 173 = 178
5 * 179 = 895 5 + 179 = 184
5 * 181 = 905 5 + 181 = 186
5 * 191 = 955 5 + 191 = 196
5 * 193 = 965 5 + 193 = 198
5 * 197 = 985 5 + 197 = 202
5 * 199 = 995 5 + 199 = 204
5 * 211 = 1055 5 + 211 = 216
5 * 223 = 1115 5 + 223 = 228
5 * 227 = 1135 5 + 227 = 232
5 * 229 = 1145 5 + 229 = 234
5 * 233 = 1165 5 + 233 = 238
5 * 239 = 1195 5 + 239 = 244
7 * 11 = 77 7 + 11 = 18
7 * 13 = 91 7 + 13 = 20
7 * 17 = 119 7 + 17 = 24
7 * 19 = 133 7 + 19 = 26
7 * 23 = 161 7 + 23 = 30
7 * 29 = 203 7 + 29 = 36
7 * 31 = 217 7 + 31 = 38
7 * 37 = 259 7 + 37 = 44
7 * 41 = 287 7 + 41 = 48
7 * 43 = 301 7 + 43 = 50
7 * 47 = 329 7 + 47 = 54
7 * 53 = 371 7 + 53 = 60
7 * 59 = 413 7 + 59 = 66
7 * 61 = 427 7 + 61 = 68
7 * 67 = 469 7 + 67 = 74
7 * 71 = 497 7 + 71 = 78
7 * 73 = 511 7 + 73 = 80
7 * 79 = 553 7 + 79 = 86
7 * 83 = 581 7 + 83 = 90
7 * 89 = 623 7 + 89 = 96
7 * 97 = 679 7 + 97 = 104
7 * 101 = 707 7 + 101 = 108
7 * 103 = 721 7 + 103 = 110
7 * 107 = 749 7 + 107 = 114
7 * 109 = 763 7 + 109 = 116
7 * 113 = 791 7 + 113 = 120
7 * 127 = 889 7 + 127 = 134
7 * 131 = 917 7 + 131 = 138
7 * 137 = 959 7 + 137 = 144
7 * 139 = 973 7 + 139 = 146
7 * 149 = 1043 7 + 149 = 156
7 * 151 = 1057 7 + 151 = 158
7 * 157 = 1099 7 + 157 = 164
7 * 163 = 1141 7 + 163 = 170
7 * 167 = 1169 7 + 167 = 174
7 * 173 = 1211 7 + 173 = 180
7 * 179 = 1253 7 + 179 = 186
7 * 181 = 1267 7 + 181 = 188
7 * 191 = 1337 7 + 191 = 198
7 * 193 = 1351 7 + 193 = 200
7 * 197 = 1379 7 + 197 = 204
7 * 199 = 1393 7 + 199 = 206
7 * 211 = 1477 7 + 211 = 218
7 * 223 = 1561 7 + 223 = 230
7 * 227 = 1589 7 + 227 = 234
7 * 229 = 1603 7 + 229 = 236
7 * 233 = 1631 7 + 233 = 240
7 * 239 = 1673 7 + 239 = 246
11 * 13 = 143 11 + 13 = 24
11 * 17 = 187 11 + 17 = 28
11 * 19 = 209 11 + 19 = 30
11 * 23 = 253 11 + 23 = 34
11 * 29 = 319 11 + 29 = 40
11 * 31 = 341 11 + 31 = 42
11 * 37 = 407 11 + 37 = 48
11 * 41 = 451 11 + 41 = 52
11 * 43 = 473 11 + 43 = 54
11 * 47 = 517 11 + 47 = 58
11 * 53 = 583 11 + 53 = 64
11 * 59 = 649 11 + 59 = 70
11 * 61 = 671 11 + 61 = 72
11 * 67 = 737 11 + 67 = 78
11 * 71 = 781 11 + 71 = 82
11 * 73 = 803 11 + 73 = 84
11 * 79 = 869 11 + 79 = 90
11 * 83 = 913 11 + 83 = 94
11 * 89 = 979 11 + 89 = 100
11 * 97 = 1067 11 + 97 = 108
11 * 101 = 1111 11 + 101 = 112
11 * 103 = 1133 11 + 103 = 114
11 * 107 = 1177 11 + 107 = 118
11 * 109 = 1199 11 + 109 = 120
11 * 113 = 1243 11 + 113 = 124
11 * 127 = 1397 11 + 127 = 138
11 * 131 = 1441 11 + 131 = 142
11 * 137 = 1507 11 + 137 = 148
11 * 139 = 1529 11 + 139 = 150
11 * 149 = 1639 11 + 149 = 160
11 * 151 = 1661 11 + 151 = 162
11 * 157 = 1727 11 + 157 = 168
11 * 163 = 1793 11 + 163 = 174
11 * 167 = 1837 11 + 167 = 178
11 * 173 = 1903 11 + 173 = 184
11 * 179 = 1969 11 + 179 = 190
11 * 181 = 1991 11 + 181 = 192
11 * 191 = 2101 11 + 191 = 202
11 * 193 = 2123 11 + 193 = 204
11 * 197 = 2167 11 + 197 = 208
11 * 199 = 2189 11 + 199 = 210
11 * 211 = 2321 11 + 211 = 222
11 * 223 = 2453 11 + 223 = 234
11 * 227 = 2497 11 + 227 = 238
11 * 229 = 2519 11 + 229 = 240
11 * 233 = 2563 11 + 233 = 244
11 * 239 = 2629 11 + 239 = 250
13 * 17 = 221 13 + 17 = 30
13 * 19 = 247 13 + 19 = 32
13 * 23 = 299 13 + 23 = 36
13 * 29 = 377 13 + 29 = 42
13 * 31 = 403 13 + 31 = 44
13 * 37 = 481 13 + 37 = 50
13 * 41 = 533 13 + 41 = 54
13 * 43 = 559 13 + 43 = 56
13 * 47 = 611 13 + 47 = 60
13 * 53 = 689 13 + 53 = 66
13 * 59 = 767 13 + 59 = 72
13 * 61 = 793 13 + 61 = 74
13 * 67 = 871 13 + 67 = 80
13 * 71 = 923 13 + 71 = 84
13 * 73 = 949 13 + 73 = 86
13 * 79 = 1027 13 + 79 = 92
13 * 83 = 1079 13 + 83 = 96
13 * 89 = 1157 13 + 89 = 102
13 * 97 = 1261 13 + 97 = 110
13 * 101 = 1313 13 + 101 = 114
13 * 103 = 1339 13 + 103 = 116
13 * 107 = 1391 13 + 107 = 120
13 * 109 = 1417 13 + 109 = 122
13 * 113 = 1469 13 + 113 = 126
13 * 127 = 1651 13 + 127 = 140
13 * 131 = 1703 13 + 131 = 144
13 * 137 = 1781 13 + 137 = 150
13 * 139 = 1807 13 + 139 = 152
13 * 149 = 1937 13 + 149 = 162
13 * 151 = 1963 13 + 151 = 164
13 * 157 = 2041 13 + 157 = 170
13 * 163 = 2119 13 + 163 = 176
13 * 167 = 2171 13 + 167 = 180
13 * 173 = 2249 13 + 173 = 186
13 * 179 = 2327 13 + 179 = 192
13 * 181 = 2353 13 + 181 = 194
13 * 191 = 2483 13 + 191 = 204
13 * 193 = 2509 13 + 193 = 206
13 * 197 = 2561 13 + 197 = 210
13 * 199 = 2587 13 + 199 = 212
13 * 211 = 2743 13 + 211 = 224
13 * 223 = 2899 13 + 223 = 236
13 * 227 = 2951 13 + 227 = 240
13 * 229 = 2977 13 + 229 = 242
13 * 233 = 3029 13 + 233 = 246
13 * 239 = 3107 13 + 239 = 252
17 * 19 = 323 17 + 19 = 36
17 * 23 = 391 17 + 23 = 40
17 * 29 = 493 17 + 29 = 46
17 * 31 = 527 17 + 31 = 48
17 * 37 = 629 17 + 37 = 54
17 * 41 = 697 17 + 41 = 58
17 * 43 = 731 17 + 43 = 60
17 * 47 = 799 17 + 47 = 64
17 * 53 = 901 17 + 53 = 70
17 * 59 = 1003 17 + 59 = 76
17 * 61 = 1037 17 + 61 = 78
17 * 67 = 1139 17 + 67 = 84
17 * 71 = 1207 17 + 71 = 88
17 * 73 = 1241 17 + 73 = 90
17 * 79 = 1343 17 + 79 = 96
17 * 83 = 1411 17 + 83 = 100
17 * 89 = 1513 17 + 89 = 106
17 * 97 = 1649 17 + 97 = 114
17 * 101 = 1717 17 + 101 = 118
17 * 103 = 1751 17 + 103 = 120
17 * 107 = 1819 17 + 107 = 124
17 * 109 = 1853 17 + 109 = 126
17 * 113 = 1921 17 + 113 = 130
17 * 127 = 2159 17 + 127 = 144
17 * 131 = 2227 17 + 131 = 148
17 * 137 = 2329 17 + 137 = 154
17 * 139 = 2363 17 + 139 = 156
17 * 149 = 2533 17 + 149 = 166
17 * 151 = 2567 17 + 151 = 168
17 * 157 = 2669 17 + 157 = 174
17 * 163 = 2771 17 + 163 = 180
17 * 167 = 2839 17 + 167 = 184
17 * 173 = 2941 17 + 173 = 190
17 * 179 = 3043 17 + 179 = 196
17 * 181 = 3077 17 + 181 = 198
17 * 191 = 3247 17 + 191 = 208
17 * 193 = 3281 17 + 193 = 210
17 * 197 = 3349 17 + 197 = 214
17 * 199 = 3383 17 + 199 = 216
17 * 211 = 3587 17 + 211 = 228
17 * 223 = 3791 17 + 223 = 240
17 * 227 = 3859 17 + 227 = 244
17 * 229 = 3893 17 + 229 = 246
17 * 233 = 3961 17 + 233 = 250
17 * 239 = 4063 17 + 239 = 256
19 * 23 = 437 19 + 23 = 42
19 * 29 = 551 19 + 29 = 48
19 * 31 = 589 19 + 31 = 50
19 * 37 = 703 19 + 37 = 56
19 * 41 = 779 19 + 41 = 60
19 * 43 = 817 19 + 43 = 62
19 * 47 = 893 19 + 47 = 66
19 * 53 = 1007 19 + 53 = 72
19 * 59 = 1121 19 + 59 = 78
19 * 61 = 1159 19 + 61 = 80
19 * 67 = 1273 19 + 67 = 86
19 * 71 = 1349 19 + 71 = 90
19 * 73 = 1387 19 + 73 = 92
19 * 79 = 1501 19 + 79 = 98
19 * 83 = 1577 19 + 83 = 102
19 * 89 = 1691 19 + 89 = 108
19 * 97 = 1843 19 + 97 = 116
19 * 101 = 1919 19 + 101 = 120
19 * 103 = 1957 19 + 103 = 122
19 * 107 = 2033 19 + 107 = 126
19 * 109 = 2071 19 + 109 = 128
19 * 113 = 2147 19 + 113 = 132
19 * 127 = 2413 19 + 127 = 146
19 * 131 = 2489 19 + 131 = 150
19 * 137 = 2603 19 + 137 = 156
19 * 139 = 2641 19 + 139 = 158
19 * 149 = 2831 19 + 149 = 168
19 * 151 = 2869 19 + 151 = 170
19 * 157 = 2983 19 + 157 = 176
19 * 163 = 3097 19 + 163 = 182
19 * 167 = 3173 19 + 167 = 186
19 * 173 = 3287 19 + 173 = 192
19 * 179 = 3401 19 + 179 = 198
19 * 181 = 3439 19 + 181 = 200
19 * 191 = 3629 19 + 191 = 210
19 * 193 = 3667 19 + 193 = 212
19 * 197 = 3743 19 + 197 = 216
19 * 199 = 3781 19 + 199 = 218
19 * 211 = 4009 19 + 211 = 230
19 * 223 = 4237 19 + 223 = 242
19 * 227 = 4313 19 + 227 = 246
19 * 229 = 4351 19 + 229 = 248
19 * 233 = 4427 19 + 233 = 252
19 * 239 = 4541 19 + 239 = 258
23 * 29 = 667 23 + 29 = 52
23 * 31 = 713 23 + 31 = 54
23 * 37 = 851 23 + 37 = 60
23 * 41 = 943 23 + 41 = 64
23 * 43 = 989 23 + 43 = 66
23 * 47 = 1081 23 + 47 = 70
23 * 53 = 1219 23 + 53 = 76
23 * 59 = 1357 23 + 59 = 82
23 * 61 = 1403 23 + 61 = 84
23 * 67 = 1541 23 + 67 = 90
23 * 71 = 1633 23 + 71 = 94
23 * 73 = 1679 23 + 73 = 96
23 * 79 = 1817 23 + 79 = 102
23 * 83 = 1909 23 + 83 = 106
23 * 89 = 2047 23 + 89 = 112
23 * 97 = 2231 23 + 97 = 120
23 * 101 = 2323 23 + 101 = 124
23 * 103 = 2369 23 + 103 = 126
23 * 107 = 2461 23 + 107 = 130
23 * 109 = 2507 23 + 109 = 132
23 * 113 = 2599 23 + 113 = 136
23 * 127 = 2921 23 + 127 = 150
23 * 131 = 3013 23 + 131 = 154
23 * 137 = 3151 23 + 137 = 160
23 * 139 = 3197 23 + 139 = 162
23 * 149 = 3427 23 + 149 = 172
23 * 151 = 3473 23 + 151 = 174
23 * 157 = 3611 23 + 157 = 180
23 * 163 = 3749 23 + 163 = 186
23 * 167 = 3841 23 + 167 = 190
23 * 173 = 3979 23 + 173 = 196
23 * 179 = 4117 23 + 179 = 202
23 * 181 = 4163 23 + 181 = 204
23 * 191 = 4393 23 + 191 = 214
23 * 193 = 4439 23 + 193 = 216
23 * 197 = 4531 23 + 197 = 220
23 * 199 = 4577 23 + 199 = 222
23 * 211 = 4853 23 + 211 = 234
23 * 223 = 5129 23 + 223 = 246
23 * 227 = 5221 23 + 227 = 250
23 * 229 = 5267 23 + 229 = 252
23 * 233 = 5359 23 + 233 = 256
23 * 239 = 5497 23 + 239 = 262
29 * 31 = 899 29 + 31 = 60
29 * 37 = 1073 29 + 37 = 66
29 * 41 = 1189 29 + 41 = 70
29 * 43 = 1247 29 + 43 = 72
29 * 47 = 1363 29 + 47 = 76
29 * 53 = 1537 29 + 53 = 82
29 * 59 = 1711 29 + 59 = 88
29 * 61 = 1769 29 + 61 = 90
29 * 67 = 1943 29 + 67 = 96
29 * 71 = 2059 29 + 71 = 100
29 * 73 = 2117 29 + 73 = 102
29 * 79 = 2291 29 + 79 = 108
29 * 83 = 2407 29 + 83 = 112
29 * 89 = 2581 29 + 89 = 118
29 * 97 = 2813 29 + 97 = 126
29 * 101 = 2929 29 + 101 = 130
29 * 103 = 2987 29 + 103 = 132
29 * 107 = 3103 29 + 107 = 136
29 * 109 = 3161 29 + 109 = 138
29 * 113 = 3277 29 + 113 = 142
29 * 127 = 3683 29 + 127 = 156
29 * 131 = 3799 29 + 131 = 160
29 * 137 = 3973 29 + 137 = 166
29 * 139 = 4031 29 + 139 = 168
29 * 149 = 4321 29 + 149 = 178
29 * 151 = 4379 29 + 151 = 180
29 * 157 = 4553 29 + 157 = 186
29 * 163 = 4727 29 + 163 = 192
29 * 167 = 4843 29 + 167 = 196
29 * 173 = 5017 29 + 173 = 202
29 * 179 = 5191 29 + 179 = 208
29 * 181 = 5249 29 + 181 = 210
29 * 191 = 5539 29 + 191 = 220
29 * 193 = 5597 29 + 193 = 222
29 * 197 = 5713 29 + 197 = 226
29 * 199 = 5771 29 + 199 = 228
29 * 211 = 6119 29 + 211 = 240
29 * 223 = 6467 29 + 223 = 252
29 * 227 = 6583 29 + 227 = 256
29 * 229 = 6641 29 + 229 = 258
29 * 233 = 6757 29 + 233 = 262
29 * 239 = 6931 29 + 239 = 268
31 * 37 = 1147 31 + 37 = 68
31 * 41 = 1271 31 + 41 = 72
31 * 43 = 1333 31 + 43 = 74
31 * 47 = 1457 31 + 47 = 78
31 * 53 = 1643 31 + 53 = 84
31 * 59 = 1829 31 + 59 = 90
31 * 61 = 1891 31 + 61 = 92
31 * 67 = 2077 31 + 67 = 98
31 * 71 = 2201 31 + 71 = 102
31 * 73 = 2263 31 + 73 = 104
31 * 79 = 2449 31 + 79 = 110
31 * 83 = 2573 31 + 83 = 114
31 * 89 = 2759 31 + 89 = 120
31 * 97 = 3007 31 + 97 = 128
31 * 101 = 3131 31 + 101 = 132
31 * 103 = 3193 31 + 103 = 134
31 * 107 = 3317 31 + 107 = 138
31 * 109 = 3379 31 + 109 = 140
31 * 113 = 3503 31 + 113 = 144
31 * 127 = 3937 31 + 127 = 158
31 * 131 = 4061 31 + 131 = 162
31 * 137 = 4247 31 + 137 = 168
31 * 139 = 4309 31 + 139 = 170
31 * 149 = 4619 31 + 149 = 180
31 * 151 = 4681 31 + 151 = 182
31 * 157 = 4867 31 + 157 = 188
31 * 163 = 5053 31 + 163 = 194
31 * 167 = 5177 31 + 167 = 198
31 * 173 = 5363 31 + 173 = 204
31 * 179 = 5549 31 + 179 = 210
31 * 181 = 5611 31 + 181 = 212
31 * 191 = 5921 31 + 191 = 222
31 * 193 = 5983 31 + 193 = 224
31 * 197 = 6107 31 + 197 = 228
31 * 199 = 6169 31 + 199 = 230
31 * 211 = 6541 31 + 211 = 242
31 * 223 = 6913 31 + 223 = 254
31 * 227 = 7037 31 + 227 = 258
31 * 229 = 7099 31 + 229 = 260
31 * 233 = 7223 31 + 233 = 264
31 * 239 = 7409 31 + 239 = 270
37 * 41 = 1517 37 + 41 = 78
37 * 43 = 1591 37 + 43 = 80
37 * 47 = 1739 37 + 47 = 84
37 * 53 = 1961 37 + 53 = 90
37 * 59 = 2183 37 + 59 = 96
37 * 61 = 2257 37 + 61 = 98
37 * 67 = 2479 37 + 67 = 104
37 * 71 = 2627 37 + 71 = 108
37 * 73 = 2701 37 + 73 = 110
37 * 79 = 2923 37 + 79 = 116
37 * 83 = 3071 37 + 83 = 120
37 * 89 = 3293 37 + 89 = 126
37 * 97 = 3589 37 + 97 = 134
37 * 101 = 3737 37 + 101 = 138
37 * 103 = 3811 37 + 103 = 140
37 * 107 = 3959 37 + 107 = 144
37 * 109 = 4033 37 + 109 = 146
37 * 113 = 4181 37 + 113 = 150
37 * 127 = 4699 37 + 127 = 164
37 * 131 = 4847 37 + 131 = 168
37 * 137 = 5069 37 + 137 = 174
37 * 139 = 5143 37 + 139 = 176
37 * 149 = 5513 37 + 149 = 186
37 * 151 = 5587 37 + 151 = 188
37 * 157 = 5809 37 + 157 = 194
37 * 163 = 6031 37 + 163 = 200
37 * 167 = 6179 37 + 167 = 204
37 * 173 = 6401 37 + 173 = 210
37 * 179 = 6623 37 + 179 = 216
37 * 181 = 6697 37 + 181 = 218
37 * 191 = 7067 37 + 191 = 228
37 * 193 = 7141 37 + 193 = 230
37 * 197 = 7289 37 + 197 = 234
37 * 199 = 7363 37 + 199 = 236
37 * 211 = 7807 37 + 211 = 248
37 * 223 = 8251 37 + 223 = 260
37 * 227 = 8399 37 + 227 = 264
37 * 229 = 8473 37 + 229 = 266
37 * 233 = 8621 37 + 233 = 270
37 * 239 = 8843 37 + 239 = 276
41 * 43 = 1763 41 + 43 = 84
41 * 47 = 1927 41 + 47 = 88
41 * 53 = 2173 41 + 53 = 94
41 * 59 = 2419 41 + 59 = 100
41 * 61 = 2501 41 + 61 = 102
41 * 67 = 2747 41 + 67 = 108
41 * 71 = 2911 41 + 71 = 112
41 * 73 = 2993 41 + 73 = 114
41 * 79 = 3239 41 + 79 = 120
41 * 83 = 3403 41 + 83 = 124
41 * 89 = 3649 41 + 89 = 130
41 * 97 = 3977 41 + 97 = 138
41 * 101 = 4141 41 + 101 = 142
41 * 103 = 4223 41 + 103 = 144
41 * 107 = 4387 41 + 107 = 148
41 * 109 = 4469 41 + 109 = 150
41 * 113 = 4633 41 + 113 = 154
41 * 127 = 5207 41 + 127 = 168
41 * 131 = 5371 41 + 131 = 172
41 * 137 = 5617 41 + 137 = 178
41 * 139 = 5699 41 + 139 = 180
41 * 149 = 6109 41 + 149 = 190
41 * 151 = 6191 41 + 151 = 192
41 * 157 = 6437 41 + 157 = 198
41 * 163 = 6683 41 + 163 = 204
41 * 167 = 6847 41 + 167 = 208
41 * 173 = 7093 41 + 173 = 214
41 * 179 = 7339 41 + 179 = 220
41 * 181 = 7421 41 + 181 = 222
41 * 191 = 7831 41 + 191 = 232
41 * 193 = 7913 41 + 193 = 234
41 * 197 = 8077 41 + 197 = 238
41 * 199 = 8159 41 + 199 = 240
41 * 211 = 8651 41 + 211 = 252
41 * 223 = 9143 41 + 223 = 264
41 * 227 = 9307 41 + 227 = 268
41 * 229 = 9389 41 + 229 = 270
41 * 233 = 9553 41 + 233 = 274
41 * 239 = 9799 41 + 239 = 280
43 * 47 = 2021 43 + 47 = 90
43 * 53 = 2279 43 + 53 = 96
43 * 59 = 2537 43 + 59 = 102
43 * 61 = 2623 43 + 61 = 104
43 * 67 = 2881 43 + 67 = 110
43 * 71 = 3053 43 + 71 = 114
43 * 73 = 3139 43 + 73 = 116
43 * 79 = 3397 43 + 79 = 122
43 * 83 = 3569 43 + 83 = 126
43 * 89 = 3827 43 + 89 = 132
43 * 97 = 4171 43 + 97 = 140
43 * 101 = 4343 43 + 101 = 144
43 * 103 = 4429 43 + 103 = 146
43 * 107 = 4601 43 + 107 = 150
43 * 109 = 4687 43 + 109 = 152
43 * 113 = 4859 43 + 113 = 156
43 * 127 = 5461 43 + 127 = 170
43 * 131 = 5633 43 + 131 = 174
43 * 137 = 5891 43 + 137 = 180
43 * 139 = 5977 43 + 139 = 182
43 * 149 = 6407 43 + 149 = 192
43 * 151 = 6493 43 + 151 = 194
43 * 157 = 6751 43 + 157 = 200
43 * 163 = 7009 43 + 163 = 206
43 * 167 = 7181 43 + 167 = 210
43 * 173 = 7439 43 + 173 = 216
43 * 179 = 7697 43 + 179 = 222
43 * 181 = 7783 43 + 181 = 224
43 * 191 = 8213 43 + 191 = 234
43 * 193 = 8299 43 + 193 = 236
43 * 197 = 8471 43 + 197 = 240
43 * 199 = 8557 43 + 199 = 242
43 * 211 = 9073 43 + 211 = 254
43 * 223 = 9589 43 + 223 = 266
43 * 227 = 9761 43 + 227 = 270
43 * 229 = 9847 43 + 229 = 272
43 * 233 = 10019 43 + 233 = 276
43 * 239 = 10277 43 + 239 = 282
47 * 53 = 2491 47 + 53 = 100
47 * 59 = 2773 47 + 59 = 106
47 * 61 = 2867 47 + 61 = 108
47 * 67 = 3149 47 + 67 = 114
47 * 71 = 3337 47 + 71 = 118
47 * 73 = 3431 47 + 73 = 120
47 * 79 = 3713 47 + 79 = 126
47 * 83 = 3901 47 + 83 = 130
47 * 89 = 4183 47 + 89 = 136
47 * 97 = 4559 47 + 97 = 144
47 * 101 = 4747 47 + 101 = 148
47 * 103 = 4841 47 + 103 = 150
47 * 107 = 5029 47 + 107 = 154
47 * 109 = 5123 47 + 109 = 156
47 * 113 = 5311 47 + 113 = 160
47 * 127 = 5969 47 + 127 = 174
47 * 131 = 6157 47 + 131 = 178
47 * 137 = 6439 47 + 137 = 184
47 * 139 = 6533 47 + 139 = 186
47 * 149 = 7003 47 + 149 = 196
47 * 151 = 7097 47 + 151 = 198
47 * 157 = 7379 47 + 157 = 204
47 * 163 = 7661 47 + 163 = 210
47 * 167 = 7849 47 + 167 = 214
47 * 173 = 8131 47 + 173 = 220
47 * 179 = 8413 47 + 179 = 226
47 * 181 = 8507 47 + 181 = 228
47 * 191 = 8977 47 + 191 = 238
47 * 193 = 9071 47 + 193 = 240
47 * 197 = 9259 47 + 197 = 244
47 * 199 = 9353 47 + 199 = 246
47 * 211 = 9917 47 + 211 = 258
47 * 223 = 10481 47 + 223 = 270
47 * 227 = 10669 47 + 227 = 274
47 * 229 = 10763 47 + 229 = 276
47 * 233 = 10951 47 + 233 = 280
47 * 239 = 11233 47 + 239 = 286
53 * 59 = 3127 53 + 59 = 112
53 * 61 = 3233 53 + 61 = 114
53 * 67 = 3551 53 + 67 = 120
53 * 71 = 3763 53 + 71 = 124
53 * 73 = 3869 53 + 73 = 126
53 * 79 = 4187 53 + 79 = 132
53 * 83 = 4399 53 + 83 = 136
53 * 89 = 4717 53 + 89 = 142
53 * 97 = 5141 53 + 97 = 150
53 * 101 = 5353 53 + 101 = 154
53 * 103 = 5459 53 + 103 = 156
53 * 107 = 5671 53 + 107 = 160
53 * 109 = 5777 53 + 109 = 162
53 * 113 = 5989 53 + 113 = 166
53 * 127 = 6731 53 + 127 = 180
53 * 131 = 6943 53 + 131 = 184
53 * 137 = 7261 53 + 137 = 190
53 * 139 = 7367 53 + 139 = 192
53 * 149 = 7897 53 + 149 = 202
53 * 151 = 8003 53 + 151 = 204
53 * 157 = 8321 53 + 157 = 210
53 * 163 = 8639 53 + 163 = 216
53 * 167 = 8851 53 + 167 = 220
53 * 173 = 9169 53 + 173 = 226
53 * 179 = 9487 53 + 179 = 232
53 * 181 = 9593 53 + 181 = 234
53 * 191 = 10123 53 + 191 = 244
53 * 193 = 10229 53 + 193 = 246
53 * 197 = 10441 53 + 197 = 250
53 * 199 = 10547 53 + 199 = 252
53 * 211 = 11183 53 + 211 = 264
53 * 223 = 11819 53 + 223 = 276
53 * 227 = 12031 53 + 227 = 280
53 * 229 = 12137 53 + 229 = 282
53 * 233 = 12349 53 + 233 = 286
53 * 239 = 12667 53 + 239 = 292
59 * 61 = 3599 59 + 61 = 120
59 * 67 = 3953 59 + 67 = 126
59 * 71 = 4189 59 + 71 = 130
59 * 73 = 4307 59 + 73 = 132
59 * 79 = 4661 59 + 79 = 138
59 * 83 = 4897 59 + 83 = 142
59 * 89 = 5251 59 + 89 = 148
59 * 97 = 5723 59 + 97 = 156
59 * 101 = 5959 59 + 101 = 160
59 * 103 = 6077 59 + 103 = 162
59 * 107 = 6313 59 + 107 = 166
59 * 109 = 6431 59 + 109 = 168
59 * 113 = 6667 59 + 113 = 172
59 * 127 = 7493 59 + 127 = 186
59 * 131 = 7729 59 + 131 = 190
59 * 137 = 8083 59 + 137 = 196
59 * 139 = 8201 59 + 139 = 198
59 * 149 = 8791 59 + 149 = 208
59 * 151 = 8909 59 + 151 = 210
59 * 157 = 9263 59 + 157 = 216
59 * 163 = 9617 59 + 163 = 222
59 * 167 = 9853 59 + 167 = 226
59 * 173 = 10207 59 + 173 = 232
59 * 179 = 10561 59 + 179 = 238
59 * 181 = 10679 59 + 181 = 240
59 * 191 = 11269 59 + 191 = 250
59 * 193 = 11387 59 + 193 = 252
59 * 197 = 11623 59 + 197 = 256
59 * 199 = 11741 59 + 199 = 258
59 * 211 = 12449 59 + 211 = 270
59 * 223 = 13157 59 + 223 = 282
59 * 227 = 13393 59 + 227 = 286
59 * 229 = 13511 59 + 229 = 288
59 * 233 = 13747 59 + 233 = 292
59 * 239 = 14101 59 + 239 = 298
61 * 67 = 4087 61 + 67 = 128
61 * 71 = 4331 61 + 71 = 132
61 * 73 = 4453 61 + 73 = 134
61 * 79 = 4819 61 + 79 = 140
61 * 83 = 5063 61 + 83 = 144
61 * 89 = 5429 61 + 89 = 150
61 * 97 = 5917 61 + 97 = 158
61 * 101 = 6161 61 + 101 = 162
61 * 103 = 6283 61 + 103 = 164
61 * 107 = 6527 61 + 107 = 168
61 * 109 = 6649 61 + 109 = 170
61 * 113 = 6893 61 + 113 = 174
61 * 127 = 7747 61 + 127 = 188
61 * 131 = 7991 61 + 131 = 192
61 * 137 = 8357 61 + 137 = 198
61 * 139 = 8479 61 + 139 = 200
61 * 149 = 9089 61 + 149 = 210
61 * 151 = 9211 61 + 151 = 212
61 * 157 = 9577 61 + 157 = 218
61 * 163 = 9943 61 + 163 = 224
61 * 167 = 10187 61 + 167 = 228
61 * 173 = 10553 61 + 173 = 234
61 * 179 = 10919 61 + 179 = 240
61 * 181 = 11041 61 + 181 = 242
61 * 191 = 11651 61 + 191 = 252
61 * 193 = 11773 61 + 193 = 254
61 * 197 = 12017 61 + 197 = 258
61 * 199 = 12139 61 + 199 = 260
61 * 211 = 12871 61 + 211 = 272
61 * 223 = 13603 61 + 223 = 284
61 * 227 = 13847 61 + 227 = 288
61 * 229 = 13969 61 + 229 = 290
61 * 233 = 14213 61 + 233 = 294
61 * 239 = 14579 61 + 239 = 300
67 * 71 = 4757 67 + 71 = 138
67 * 73 = 4891 67 + 73 = 140
67 * 79 = 5293 67 + 79 = 146
67 * 83 = 5561 67 + 83 = 150
67 * 89 = 5963 67 + 89 = 156
67 * 97 = 6499 67 + 97 = 164
67 * 101 = 6767 67 + 101 = 168
67 * 103 = 6901 67 + 103 = 170
67 * 107 = 7169 67 + 107 = 174
67 * 109 = 7303 67 + 109 = 176
67 * 113 = 7571 67 + 113 = 180
67 * 127 = 8509 67 + 127 = 194
67 * 131 = 8777 67 + 131 = 198
67 * 137 = 9179 67 + 137 = 204
67 * 139 = 9313 67 + 139 = 206
67 * 149 = 9983 67 + 149 = 216
67 * 151 = 10117 67 + 151 = 218
67 * 157 = 10519 67 + 157 = 224
67 * 163 = 10921 67 + 163 = 230
67 * 167 = 11189 67 + 167 = 234
67 * 173 = 11591 67 + 173 = 240
67 * 179 = 11993 67 + 179 = 246
67 * 181 = 12127 67 + 181 = 248
67 * 191 = 12797 67 + 191 = 258
67 * 193 = 12931 67 + 193 = 260
67 * 197 = 13199 67 + 197 = 264
67 * 199 = 13333 67 + 199 = 266
67 * 211 = 14137 67 + 211 = 278
67 * 223 = 14941 67 + 223 = 290
67 * 227 = 15209 67 + 227 = 294
67 * 229 = 15343 67 + 229 = 296
67 * 233 = 15611 67 + 233 = 300
67 * 239 = 16013 67 + 239 = 306
71 * 73 = 5183 71 + 73 = 144
71 * 79 = 5609 71 + 79 = 150
71 * 83 = 5893 71 + 83 = 154
71 * 89 = 6319 71 + 89 = 160
71 * 97 = 6887 71 + 97 = 168
71 * 101 = 7171 71 + 101 = 172
71 * 103 = 7313 71 + 103 = 174
71 * 107 = 7597 71 + 107 = 178
71 * 109 = 7739 71 + 109 = 180
71 * 113 = 8023 71 + 113 = 184
71 * 127 = 9017 71 + 127 = 198
71 * 131 = 9301 71 + 131 = 202
71 * 137 = 9727 71 + 137 = 208
71 * 139 = 9869 71 + 139 = 210
71 * 149 = 10579 71 + 149 = 220
71 * 151 = 10721 71 + 151 = 222
71 * 157 = 11147 71 + 157 = 228
71 * 163 = 11573 71 + 163 = 234
71 * 167 = 11857 71 + 167 = 238
71 * 173 = 12283 71 + 173 = 244
71 * 179 = 12709 71 + 179 = 250
71 * 181 = 12851 71 + 181 = 252
71 * 191 = 13561 71 + 191 = 262
71 * 193 = 13703 71 + 193 = 264
71 * 197 = 13987 71 + 197 = 268
71 * 199 = 14129 71 + 199 = 270
71 * 211 = 14981 71 + 211 = 282
71 * 223 = 15833 71 + 223 = 294
71 * 227 = 16117 71 + 227 = 298
71 * 229 = 16259 71 + 229 = 300
71 * 233 = 16543 71 + 233 = 304
71 * 239 = 16969 71 + 239 = 310
73 * 79 = 5767 73 + 79 = 152
73 * 83 = 6059 73 + 83 = 156
73 * 89 = 6497 73 + 89 = 162
73 * 97 = 7081 73 + 97 = 170
73 * 101 = 7373 73 + 101 = 174
73 * 103 = 7519 73 + 103 = 176
73 * 107 = 7811 73 + 107 = 180
73 * 109 = 7957 73 + 109 = 182
73 * 113 = 8249 73 + 113 = 186
73 * 127 = 9271 73 + 127 = 200
73 * 131 = 9563 73 + 131 = 204
73 * 137 = 10001 73 + 137 = 210
73 * 139 = 10147 73 + 139 = 212
73 * 149 = 10877 73 + 149 = 222
73 * 151 = 11023 73 + 151 = 224
73 * 157 = 11461 73 + 157 = 230
73 * 163 = 11899 73 + 163 = 236
73 * 167 = 12191 73 + 167 = 240
73 * 173 = 12629 73 + 173 = 246
73 * 179 = 13067 73 + 179 = 252
73 * 181 = 13213 73 + 181 = 254
73 * 191 = 13943 73 + 191 = 264
73 * 193 = 14089 73 + 193 = 266
73 * 197 = 14381 73 + 197 = 270
73 * 199 = 14527 73 + 199 = 272
73 * 211 = 15403 73 + 211 = 284
73 * 223 = 16279 73 + 223 = 296
73 * 227 = 16571 73 + 227 = 300
73 * 229 = 16717 73 + 229 = 302
73 * 233 = 17009 73 + 233 = 306
73 * 239 = 17447 73 + 239 = 312
79 * 83 = 6557 79 + 83 = 162
79 * 89 = 7031 79 + 89 = 168
79 * 97 = 7663 79 + 97 = 176
79 * 101 = 7979 79 + 101 = 180
79 * 103 = 8137 79 + 103 = 182
79 * 107 = 8453 79 + 107 = 186
79 * 109 = 8611 79 + 109 = 188
79 * 113 = 8927 79 + 113 = 192
79 * 127 = 10033 79 + 127 = 206
79 * 131 = 10349 79 + 131 = 210
79 * 137 = 10823 79 + 137 = 216
79 * 139 = 10981 79 + 139 = 218
79 * 149 = 11771 79 + 149 = 228
79 * 151 = 11929 79 + 151 = 230
79 * 157 = 12403 79 + 157 = 236
79 * 163 = 12877 79 + 163 = 242
79 * 167 = 13193 79 + 167 = 246
79 * 173 = 13667 79 + 173 = 252
79 * 179 = 14141 79 + 179 = 258
79 * 181 = 14299 79 + 181 = 260
79 * 191 = 15089 79 + 191 = 270
79 * 193 = 15247 79 + 193 = 272
79 * 197 = 15563 79 + 197 = 276
79 * 199 = 15721 79 + 199 = 278
79 * 211 = 16669 79 + 211 = 290
79 * 223 = 17617 79 + 223 = 302
79 * 227 = 17933 79 + 227 = 306
79 * 229 = 18091 79 + 229 = 308
79 * 233 = 18407 79 + 233 = 312
79 * 239 = 18881 79 + 239 = 318
83 * 89 = 7387 83 + 89 = 172
83 * 97 = 8051 83 + 97 = 180
83 * 101 = 8383 83 + 101 = 184
83 * 103 = 8549 83 + 103 = 186
83 * 107 = 8881 83 + 107 = 190
83 * 109 = 9047 83 + 109 = 192
83 * 113 = 9379 83 + 113 = 196
83 * 127 = 10541 83 + 127 = 210
83 * 131 = 10873 83 + 131 = 214
83 * 137 = 11371 83 + 137 = 220
83 * 139 = 11537 83 + 139 = 222
83 * 149 = 12367 83 + 149 = 232
83 * 151 = 12533 83 + 151 = 234
83 * 157 = 13031 83 + 157 = 240
83 * 163 = 13529 83 + 163 = 246
83 * 167 = 13861 83 + 167 = 250
83 * 173 = 14359 83 + 173 = 256
83 * 179 = 14857 83 + 179 = 262
83 * 181 = 15023 83 + 181 = 264
83 * 191 = 15853 83 + 191 = 274
83 * 193 = 16019 83 + 193 = 276
83 * 197 = 16351 83 + 197 = 280
83 * 199 = 16517 83 + 199 = 282
83 * 211 = 17513 83 + 211 = 294
83 * 223 = 18509 83 + 223 = 306
83 * 227 = 18841 83 + 227 = 310
83 * 229 = 19007 83 + 229 = 312
83 * 233 = 19339 83 + 233 = 316
83 * 239 = 19837 83 + 239 = 322
89 * 97 = 8633 89 + 97 = 186
89 * 101 = 8989 89 + 101 = 190
89 * 103 = 9167 89 + 103 = 192
89 * 107 = 9523 89 + 107 = 196
89 * 109 = 9701 89 + 109 = 198
89 * 113 = 10057 89 + 113 = 202
89 * 127 = 11303 89 + 127 = 216
89 * 131 = 11659 89 + 131 = 220
89 * 137 = 12193 89 + 137 = 226
89 * 139 = 12371 89 + 139 = 228
89 * 149 = 13261 89 + 149 = 238
89 * 151 = 13439 89 + 151 = 240
89 * 157 = 13973 89 + 157 = 246
89 * 163 = 14507 89 + 163 = 252
89 * 167 = 14863 89 + 167 = 256
89 * 173 = 15397 89 + 173 = 262
89 * 179 = 15931 89 + 179 = 268
89 * 181 = 16109 89 + 181 = 270
89 * 191 = 16999 89 + 191 = 280
89 * 193 = 17177 89 + 193 = 282
89 * 197 = 17533 89 + 197 = 286
89 * 199 = 17711 89 + 199 = 288
89 * 211 = 18779 89 + 211 = 300
89 * 223 = 19847 89 + 223 = 312
89 * 227 = 20203 89 + 227 = 316
89 * 229 = 20381 89 + 229 = 318
89 * 233 = 20737 89 + 233 = 322
89 * 239 = 21271 89 + 239 = 328
97 * 101 = 9797 97 + 101 = 198
97 * 103 = 9991 97 + 103 = 200
97 * 107 = 10379 97 + 107 = 204
97 * 109 = 10573 97 + 109 = 206
97 * 113 = 10961 97 + 113 = 210
97 * 127 = 12319 97 + 127 = 224
97 * 131 = 12707 97 + 131 = 228
97 * 137 = 13289 97 + 137 = 234
97 * 139 = 13483 97 + 139 = 236
97 * 149 = 14453 97 + 149 = 246
97 * 151 = 14647 97 + 151 = 248
97 * 157 = 15229 97 + 157 = 254
97 * 163 = 15811 97 + 163 = 260
97 * 167 = 16199 97 + 167 = 264
97 * 173 = 16781 97 + 173 = 270
97 * 179 = 17363 97 + 179 = 276
97 * 181 = 17557 97 + 181 = 278
97 * 191 = 18527 97 + 191 = 288
97 * 193 = 18721 97 + 193 = 290
97 * 197 = 19109 97 + 197 = 294
97 * 199 = 19303 97 + 199 = 296
97 * 211 = 20467 97 + 211 = 308
97 * 223 = 21631 97 + 223 = 320
97 * 227 = 22019 97 + 227 = 324
97 * 229 = 22213 97 + 229 = 326
97 * 233 = 22601 97 + 233 = 330
97 * 239 = 23183 97 + 239 = 336
101 * 103 = 10403 101 + 103 = 204
101 * 107 = 10807 101 + 107 = 208
101 * 109 = 11009 101 + 109 = 210
101 * 113 = 11413 101 + 113 = 214
101 * 127 = 12827 101 + 127 = 228
101 * 131 = 13231 101 + 131 = 232
101 * 137 = 13837 101 + 137 = 238
101 * 139 = 14039 101 + 139 = 240
101 * 149 = 15049 101 + 149 = 250
101 * 151 = 15251 101 + 151 = 252
101 * 157 = 15857 101 + 157 = 258
101 * 163 = 16463 101 + 163 = 264
101 * 167 = 16867 101 + 167 = 268
101 * 173 = 17473 101 + 173 = 274
101 * 179 = 18079 101 + 179 = 280
101 * 181 = 18281 101 + 181 = 282
101 * 191 = 19291 101 + 191 = 292
101 * 193 = 19493 101 + 193 = 294
101 * 197 = 19897 101 + 197 = 298
101 * 199 = 20099 101 + 199 = 300
101 * 211 = 21311 101 + 211 = 312
101 * 223 = 22523 101 + 223 = 324
101 * 227 = 22927 101 + 227 = 328
101 * 229 = 23129 101 + 229 = 330
101 * 233 = 23533 101 + 233 = 334
101 * 239 = 24139 101 + 239 = 340
103 * 107 = 11021 103 + 107 = 210
103 * 109 = 11227 103 + 109 = 212
103 * 113 = 11639 103 + 113 = 216
103 * 127 = 13081 103 + 127 = 230
103 * 131 = 13493 103 + 131 = 234
103 * 137 = 14111 103 + 137 = 240
103 * 139 = 14317 103 + 139 = 242
103 * 149 = 15347 103 + 149 = 252
103 * 151 = 15553 103 + 151 = 254
103 * 157 = 16171 103 + 157 = 260
103 * 163 = 16789 103 + 163 = 266
103 * 167 = 17201 103 + 167 = 270
103 * 173 = 17819 103 + 173 = 276
103 * 179 = 18437 103 + 179 = 282
103 * 181 = 18643 103 + 181 = 284
103 * 191 = 19673 103 + 191 = 294
103 * 193 = 19879 103 + 193 = 296
103 * 197 = 20291 103 + 197 = 300
103 * 199 = 20497 103 + 199 = 302
103 * 211 = 21733 103 + 211 = 314
103 * 223 = 22969 103 + 223 = 326
103 * 227 = 23381 103 + 227 = 330
103 * 229 = 23587 103 + 229 = 332
103 * 233 = 23999 103 + 233 = 336
103 * 239 = 24617 103 + 239 = 342
107 * 109 = 11663 107 + 109 = 216
107 * 113 = 12091 107 + 113 = 220
107 * 127 = 13589 107 + 127 = 234
107 * 131 = 14017 107 + 131 = 238
107 * 137 = 14659 107 + 137 = 244
107 * 139 = 14873 107 + 139 = 246
107 * 149 = 15943 107 + 149 = 256
107 * 151 = 16157 107 + 151 = 258
107 * 157 = 16799 107 + 157 = 264
107 * 163 = 17441 107 + 163 = 270
107 * 167 = 17869 107 + 167 = 274
107 * 173 = 18511 107 + 173 = 280
107 * 179 = 19153 107 + 179 = 286
107 * 181 = 19367 107 + 181 = 288
107 * 191 = 20437 107 + 191 = 298
107 * 193 = 20651 107 + 193 = 300
107 * 197 = 21079 107 + 197 = 304
107 * 199 = 21293 107 + 199 = 306
107 * 211 = 22577 107 + 211 = 318
107 * 223 = 23861 107 + 223 = 330
107 * 227 = 24289 107 + 227 = 334
107 * 229 = 24503 107 + 229 = 336
107 * 233 = 24931 107 + 233 = 340
107 * 239 = 25573 107 + 239 = 346
109 * 113 = 12317 109 + 113 = 222
109 * 127 = 13843 109 + 127 = 236
109 * 131 = 14279 109 + 131 = 240
109 * 137 = 14933 109 + 137 = 246
109 * 139 = 15151 109 + 139 = 248
109 * 149 = 16241 109 + 149 = 258
109 * 151 = 16459 109 + 151 = 260
109 * 157 = 17113 109 + 157 = 266
109 * 163 = 17767 109 + 163 = 272
109 * 167 = 18203 109 + 167 = 276
109 * 173 = 18857 109 + 173 = 282
109 * 179 = 19511 109 + 179 = 288
109 * 181 = 19729 109 + 181 = 290
109 * 191 = 20819 109 + 191 = 300
109 * 193 = 21037 109 + 193 = 302
109 * 197 = 21473 109 + 197 = 306
109 * 199 = 21691 109 + 199 = 308
109 * 211 = 22999 109 + 211 = 320
109 * 223 = 24307 109 + 223 = 332
109 * 227 = 24743 109 + 227 = 336
109 * 229 = 24961 109 + 229 = 338
109 * 233 = 25397 109 + 233 = 342
109 * 239 = 26051 109 + 239 = 348
113 * 127 = 14351 113 + 127 = 240
113 * 131 = 14803 113 + 131 = 244
113 * 137 = 15481 113 + 137 = 250
113 * 139 = 15707 113 + 139 = 252
113 * 149 = 16837 113 + 149 = 262
113 * 151 = 17063 113 + 151 = 264
113 * 157 = 17741 113 + 157 = 270
113 * 163 = 18419 113 + 163 = 276
113 * 167 = 18871 113 + 167 = 280
113 * 173 = 19549 113 + 173 = 286
113 * 179 = 20227 113 + 179 = 292
113 * 181 = 20453 113 + 181 = 294
113 * 191 = 21583 113 + 191 = 304
113 * 193 = 21809 113 + 193 = 306
113 * 197 = 22261 113 + 197 = 310
113 * 199 = 22487 113 + 199 = 312
113 * 211 = 23843 113 + 211 = 324
113 * 223 = 25199 113 + 223 = 336
113 * 227 = 25651 113 + 227 = 340
113 * 229 = 25877 113 + 229 = 342
113 * 233 = 26329 113 + 233 = 346
113 * 239 = 27007 113 + 239 = 352
127 * 131 = 16637 127 + 131 = 258
127 * 137 = 17399 127 + 137 = 264
127 * 139 = 17653 127 + 139 = 266
127 * 149 = 18923 127 + 149 = 276
127 * 151 = 19177 127 + 151 = 278
127 * 157 = 19939 127 + 157 = 284
127 * 163 = 20701 127 + 163 = 290
127 * 167 = 21209 127 + 167 = 294
127 * 173 = 21971 127 + 173 = 300
127 * 179 = 22733 127 + 179 = 306
127 * 181 = 22987 127 + 181 = 308
127 * 191 = 24257 127 + 191 = 318
127 * 193 = 24511 127 + 193 = 320
127 * 197 = 25019 127 + 197 = 324
127 * 199 = 25273 127 + 199 = 326
127 * 211 = 26797 127 + 211 = 338
127 * 223 = 28321 127 + 223 = 350
127 * 227 = 28829 127 + 227 = 354
127 * 229 = 29083 127 + 229 = 356
127 * 233 = 29591 127 + 233 = 360
127 * 239 = 30353 127 + 239 = 366
131 * 137 = 17947 131 + 137 = 268
131 * 139 = 18209 131 + 139 = 270
131 * 149 = 19519 131 + 149 = 280
131 * 151 = 19781 131 + 151 = 282
131 * 157 = 20567 131 + 157 = 288
131 * 163 = 21353 131 + 163 = 294
131 * 167 = 21877 131 + 167 = 298
131 * 173 = 22663 131 + 173 = 304
131 * 179 = 23449 131 + 179 = 310
131 * 181 = 23711 131 + 181 = 312
131 * 191 = 25021 131 + 191 = 322
131 * 193 = 25283 131 + 193 = 324
131 * 197 = 25807 131 + 197 = 328
131 * 199 = 26069 131 + 199 = 330
131 * 211 = 27641 131 + 211 = 342
131 * 223 = 29213 131 + 223 = 354
131 * 227 = 29737 131 + 227 = 358
131 * 229 = 29999 131 + 229 = 360
131 * 233 = 30523 131 + 233 = 364
131 * 239 = 31309 131 + 239 = 370
137 * 139 = 19043 137 + 139 = 276
137 * 149 = 20413 137 + 149 = 286
137 * 151 = 20687 137 + 151 = 288
137 * 157 = 21509 137 + 157 = 294
137 * 163 = 22331 137 + 163 = 300
137 * 167 = 22879 137 + 167 = 304
137 * 173 = 23701 137 + 173 = 310
137 * 179 = 24523 137 + 179 = 316
137 * 181 = 24797 137 + 181 = 318
137 * 191 = 26167 137 + 191 = 328
137 * 193 = 26441 137 + 193 = 330
137 * 197 = 26989 137 + 197 = 334
137 * 199 = 27263 137 + 199 = 336
137 * 211 = 28907 137 + 211 = 348
137 * 223 = 30551 137 + 223 = 360
137 * 227 = 31099 137 + 227 = 364
137 * 229 = 31373 137 + 229 = 366
137 * 233 = 31921 137 + 233 = 370
137 * 239 = 32743 137 + 239 = 376
139 * 149 = 20711 139 + 149 = 288
139 * 151 = 20989 139 + 151 = 290
139 * 157 = 21823 139 + 157 = 296
139 * 163 = 22657 139 + 163 = 302
139 * 167 = 23213 139 + 167 = 306
139 * 173 = 24047 139 + 173 = 312
139 * 179 = 24881 139 + 179 = 318
139 * 181 = 25159 139 + 181 = 320
139 * 191 = 26549 139 + 191 = 330
139 * 193 = 26827 139 + 193 = 332
139 * 197 = 27383 139 + 197 = 336
139 * 199 = 27661 139 + 199 = 338
139 * 211 = 29329 139 + 211 = 350
139 * 223 = 30997 139 + 223 = 362
139 * 227 = 31553 139 + 227 = 366
139 * 229 = 31831 139 + 229 = 368
139 * 233 = 32387 139 + 233 = 372
139 * 239 = 33221 139 + 239 = 378
149 * 151 = 22499 149 + 151 = 300
149 * 157 = 23393 149 + 157 = 306
149 * 163 = 24287 149 + 163 = 312
149 * 167 = 24883 149 + 167 = 316
149 * 173 = 25777 149 + 173 = 322
149 * 179 = 26671 149 + 179 = 328
149 * 181 = 26969 149 + 181 = 330
149 * 191 = 28459 149 + 191 = 340
149 * 193 = 28757 149 + 193 = 342
149 * 197 = 29353 149 + 197 = 346
149 * 199 = 29651 149 + 199 = 348
149 * 211 = 31439 149 + 211 = 360
149 * 223 = 33227 149 + 223 = 372
149 * 227 = 33823 149 + 227 = 376
149 * 229 = 34121 149 + 229 = 378
149 * 233 = 34717 149 + 233 = 382
149 * 239 = 35611 149 + 239 = 388
151 * 157 = 23707 151 + 157 = 308
151 * 163 = 24613 151 + 163 = 314
151 * 167 = 25217 151 + 167 = 318
151 * 173 = 26123 151 + 173 = 324
151 * 179 = 27029 151 + 179 = 330
151 * 181 = 27331 151 + 181 = 332
151 * 191 = 28841 151 + 191 = 342
151 * 193 = 29143 151 + 193 = 344
151 * 197 = 29747 151 + 197 = 348
151 * 199 = 30049 151 + 199 = 350
151 * 211 = 31861 151 + 211 = 362
151 * 223 = 33673 151 + 223 = 374
151 * 227 = 34277 151 + 227 = 378
151 * 229 = 34579 151 + 229 = 380
151 * 233 = 35183 151 + 233 = 384
151 * 239 = 36089 151 + 239 = 390
157 * 163 = 25591 157 + 163 = 320
157 * 167 = 26219 157 + 167 = 324
157 * 173 = 27161 157 + 173 = 330
157 * 179 = 28103 157 + 179 = 336
157 * 181 = 28417 157 + 181 = 338
157 * 191 = 29987 157 + 191 = 348
157 * 193 = 30301 157 + 193 = 350
157 * 197 = 30929 157 + 197 = 354
157 * 199 = 31243 157 + 199 = 356
157 * 211 = 33127 157 + 211 = 368
157 * 223 = 35011 157 + 223 = 380
157 * 227 = 35639 157 + 227 = 384
157 * 229 = 35953 157 + 229 = 386
157 * 233 = 36581 157 + 233 = 390
157 * 239 = 37523 157 + 239 = 396
163 * 167 = 27221 163 + 167 = 330
163 * 173 = 28199 163 + 173 = 336
163 * 179 = 29177 163 + 179 = 342
163 * 181 = 29503 163 + 181 = 344
163 * 191 = 31133 163 + 191 = 354
163 * 193 = 31459 163 + 193 = 356
163 * 197 = 32111 163 + 197 = 360
163 * 199 = 32437 163 + 199 = 362
163 * 211 = 34393 163 + 211 = 374
163 * 223 = 36349 163 + 223 = 386
163 * 227 = 37001 163 + 227 = 390
163 * 229 = 37327 163 + 229 = 392
163 * 233 = 37979 163 + 233 = 396
163 * 239 = 38957 163 + 239 = 402
167 * 173 = 28891 167 + 173 = 340
167 * 179 = 29893 167 + 179 = 346
167 * 181 = 30227 167 + 181 = 348
167 * 191 = 31897 167 + 191 = 358
167 * 193 = 32231 167 + 193 = 360
167 * 197 = 32899 167 + 197 = 364
167 * 199 = 33233 167 + 199 = 366
167 * 211 = 35237 167 + 211 = 378
167 * 223 = 37241 167 + 223 = 390
167 * 227 = 37909 167 + 227 = 394
167 * 229 = 38243 167 + 229 = 396
167 * 233 = 38911 167 + 233 = 400
167 * 239 = 39913 167 + 239 = 406
173 * 179 = 30967 173 + 179 = 352
173 * 181 = 31313 173 + 181 = 354
173 * 191 = 33043 173 + 191 = 364
173 * 193 = 33389 173 + 193 = 366
173 * 197 = 34081 173 + 197 = 370
173 * 199 = 34427 173 + 199 = 372
173 * 211 = 36503 173 + 211 = 384
173 * 223 = 38579 173 + 223 = 396
173 * 227 = 39271 173 + 227 = 400
173 * 229 = 39617 173 + 229 = 402
173 * 233 = 40309 173 + 233 = 406
173 * 239 = 41347 173 + 239 = 412
179 * 181 = 32399 179 + 181 = 360
179 * 191 = 34189 179 + 191 = 370
179 * 193 = 34547 179 + 193 = 372
179 * 197 = 35263 179 + 197 = 376
179 * 199 = 35621 179 + 199 = 378
179 * 211 = 37769 179 + 211 = 390
179 * 223 = 39917 179 + 223 = 402
179 * 227 = 40633 179 + 227 = 406
179 * 229 = 40991 179 + 229 = 408
179 * 233 = 41707 179 + 233 = 412
179 * 239 = 42781 179 + 239 = 418
181 * 191 = 34571 181 + 191 = 372
181 * 193 = 34933 181 + 193 = 374
181 * 197 = 35657 181 + 197 = 378
181 * 199 = 36019 181 + 199 = 380
181 * 211 = 38191 181 + 211 = 392
181 * 223 = 40363 181 + 223 = 404
181 * 227 = 41087 181 + 227 = 408
181 * 229 = 41449 181 + 229 = 410
181 * 233 = 42173 181 + 233 = 414
181 * 239 = 43259 181 + 239 = 420
191 * 193 = 36863 191 + 193 = 384
191 * 197 = 37627 191 + 197 = 388
191 * 199 = 38009 191 + 199 = 390
191 * 211 = 40301 191 + 211 = 402
191 * 223 = 42593 191 + 223 = 414
191 * 227 = 43357 191 + 227 = 418
191 * 229 = 43739 191 + 229 = 420
191 * 233 = 44503 191 + 233 = 424
191 * 239 = 45649 191 + 239 = 430
193 * 197 = 38021 193 + 197 = 390
193 * 199 = 38407 193 + 199 = 392
193 * 211 = 40723 193 + 211 = 404
193 * 223 = 43039 193 + 223 = 416
193 * 227 = 43811 193 + 227 = 420
193 * 229 = 44197 193 + 229 = 422
193 * 233 = 44969 193 + 233 = 426
193 * 239 = 46127 193 + 239 = 432
197 * 199 = 39203 197 + 199 = 396
197 * 211 = 41567 197 + 211 = 408
197 * 223 = 43931 197 + 223 = 420
197 * 227 = 44719 197 + 227 = 424
197 * 229 = 45113 197 + 229 = 426
197 * 233 = 45901 197 + 233 = 430
197 * 239 = 47083 197 + 239 = 436
199 * 211 = 41989 199 + 211 = 410
199 * 223 = 44377 199 + 223 = 422
199 * 227 = 45173 199 + 227 = 426
199 * 229 = 45571 199 + 229 = 428
199 * 233 = 46367 199 + 233 = 432
199 * 239 = 47561 199 + 239 = 438
211 * 223 = 47053 211 + 223 = 434
211 * 227 = 47897 211 + 227 = 438
211 * 229 = 48319 211 + 229 = 440
211 * 233 = 49163 211 + 233 = 444
211 * 239 = 50429 211 + 239 = 450
223 * 227 = 50621 223 + 227 = 450
223 * 229 = 51067 223 + 229 = 452
223 * 233 = 51959 223 + 233 = 456
223 * 239 = 53297 223 + 239 = 462
227 * 229 = 51983 227 + 229 = 456
227 * 233 = 52891 227 + 233 = 460
227 * 239 = 54253 227 + 239 = 466
229 * 233 = 53357 229 + 233 = 462
229 * 239 = 54731 229 + 239 = 468
233 * 239 = 55687 233 + 239 = 472
Vinicius Claudino Ferraz
2017-08-11 18:51:46 UTC
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This is topology.
M = {6,8,10,12,...} is the space.
it's covered by B(p, q) = {p + q; p is prime, q is prime} balls several times
and we need a basis. A disjoint coberture.
Archimedes Plutonium
2017-08-11 21:20:54 UTC
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Post by Archimedes Plutonium
In these columns of 2byN matrices, I propose that they alone are all the axioms that embody
(snipped)

Alright, what I was saying is that this Schemata is containing all the All the Axioms embodied in the subject of both Arithmetic and Algebra. If you need a axiom for a proof, and if you find that feature trait in this schemata, then the schemata justifies your step of the proof.

Arithmetic and Algebra schemata of all its axioms


0 1

0 2
1 1


0 3
1 2

0 4
1 3
2 2


0 5
1 4
2 3


0 6
1 5
2 4
3 3

0 7
1 6
2 5
3 4


0 8
1 7
2 6
3 5
4 4

0 9
1 8
2 7
3 6
4 5


0 10
1 9
2 8
3 7
4 6
5 5


0 11
1 10
2 9
3 8
4 7
5 6


0 12
1 11
2 10
3 9
4 8
5 7
6 6

Now, let us see how Mathematical Induction is a feature of that schemata.
Archimedes Plutonium
2017-08-11 22:21:56 UTC
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Can we see how given N=10 it implies N=11, or given N=11 implies N=12.

AP
Archimedes Plutonium
2017-08-12 05:02:08 UTC
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Post by Archimedes Plutonium
Can we see how given N=10 it implies N=11, or given N=11 implies N=12.
So what is Mathematical Induction, really? Is it not the idea that if you have a property true for 1 and if you chose any other large number N, assume the property holds for N, and then show that the property holds for N+1, that what you have are all the Counting Numbers and all the Counting Numbers hold true for the property.

I like to think of Math Induction as another method of crafting the Counting Numbers. The first method is to start with 1 and keep adding 1 to it building 2, 3, 4, 5, etc. A second method is you have 1, now you assume another number N is a counting number, and if you show that its neighbor N+1 is a counting number, well, you have shown these numbers are Counting numbers. And there is no hole or gap, because N is "in general". N can be 2 or some huge number. N can be 3, or 4, or 5, or any number and if its neighbor N+1 is a counting number, then you have the set of all counting numbers.

So, to me, Mathematical Induction is an alternative means of building the Counting Numbers.

So do the Matrix Columns have the Mathematical Induction? Do the Columns suggest that given an N, it implies a N+1.

Well, I am certain the Columns obey the Successor function method of building the Counting Numbers, as you can see, each higher number is added a 1 from previous number. So if the Columns display a building of the Counting Numbers by adding 1, and the Successor function is a alternate method of building from Mathematical Induction, means the Columns have Mathematical Induction embedded in them.

And, maybe the Columns can explain Math Induction, far better than all words of the past has done in explaining. Here, the explaining of Math Induction is enhanced as meaning that just because you have a column for N, you can always increase that column to N+1, regardless if N is odd or even.

Arithmetic and Algebra schemata of all its axioms


0 1

0 2
1 1


0 3
1 2

0 4
1 3
2 2


0 5
1 4
2 3


0 6
1 5
2 4
3 3

0 7
1 6
2 5
3 4


0 8
1 7
2 6
3 5
4 4

0 9
1 8
2 7
3 6
4 5


0 10
1 9
2 8
3 7
4 6
5 5


0 11
1 10
2 9
3 8
4 7
5 6


0 12
1 11
2 10
3 9
4 8
5 7
6 6

AP
b***@gmail.com
2017-08-12 09:59:13 UTC
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Again to much crystal meth? Freely mixing things you have
heard such Kepler conjecture, now there are suddently

axioms packed. AP brain farto, as usually haluzinating over
his own nonsense, not a single line of math for 30 years,

only spamming sci.math with his imbecil drivel.
Post by Archimedes Plutonium
Post by Archimedes Plutonium
Can we see how given N=10 it implies N=11, or given N=11 implies N=12.
So what is Mathematical Induction, really? Is it not the idea that if you have a property true for 1 and if you chose any other large number N, assume the property holds for N, and then show that the property holds for N+1, that what you have are all the Counting Numbers and all the Counting Numbers hold true for the property.
I like to think of Math Induction as another method of crafting the Counting Numbers. The first method is to start with 1 and keep adding 1 to it building 2, 3, 4, 5, etc. A second method is you have 1, now you assume another number N is a counting number, and if you show that its neighbor N+1 is a counting number, well, you have shown these numbers are Counting numbers. And there is no hole or gap, because N is "in general". N can be 2 or some huge number. N can be 3, or 4, or 5, or any number and if its neighbor N+1 is a counting number, then you have the set of all counting numbers.
So, to me, Mathematical Induction is an alternative means of building the Counting Numbers.
So do the Matrix Columns have the Mathematical Induction? Do the Columns suggest that given an N, it implies a N+1.
Well, I am certain the Columns obey the Successor function method of building the Counting Numbers, as you can see, each higher number is added a 1 from previous number. So if the Columns display a building of the Counting Numbers by adding 1, and the Successor function is a alternate method of building from Mathematical Induction, means the Columns have Mathematical Induction embedded in them.
And, maybe the Columns can explain Math Induction, far better than all words of the past has done in explaining. Here, the explaining of Math Induction is enhanced as meaning that just because you have a column for N, you can always increase that column to N+1, regardless if N is odd or even.
Arithmetic and Algebra schemata of all its axioms
0 1
0 2
1 1
0 3
1 2
0 4
1 3
2 2
0 5
1 4
2 3
0 6
1 5
2 4
3 3
0 7
1 6
2 5
3 4
0 8
1 7
2 6
3 5
4 4
0 9
1 8
2 7
3 6
4 5
0 10
1 9
2 8
3 7
4 6
5 5
0 11
1 10
2 9
3 8
4 7
5 6
0 12
1 11
2 10
3 9
4 8
5 7
6 6
AP
Archimedes Plutonium
2017-08-12 10:24:58 UTC
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Now in the proof of Goldbach we recognize that the lined up Bertrand primes in each column such as 3+5 with 3*5 in the 8 column are complements, 8 complement to 15. So we have to ask whether the Goldbach is the only spot in mathematics where complements of a+b with a*b comes to prominence. Surely there must be other topics in math where complements serve a major role.

AP
b***@gmail.com
2017-08-12 11:58:06 UTC
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Just name dropping and complete nonsense.
As usual not a single line of math by AP brain farto.
Post by Archimedes Plutonium
Now in the proof of Goldbach we recognize that the lined up Bertrand primes in each column such as 3+5 with 3*5 in the 8 column are complements, 8 complement to 15. So we have to ask whether the Goldbach is the only spot in mathematics where complements of a+b with a*b comes to prominence. Surely there must be other topics in math where complements serve a major role.
AP
Archimedes Plutonium
2017-08-12 18:19:30 UTC
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Post by Archimedes Plutonium
Now in the proof of Goldbach we recognize that the lined up Bertrand primes in each column such as 3+5 with 3*5 in the 8 column are complements, 8 complement to 15. So we have to ask whether the Goldbach is the only spot in mathematics where complements of a+b with a*b comes to prominence. Surely there must be other topics in math where complements serve a major role.
Alright, I made a beautiful breakthrough in mathematics of the Peano Axioms. And I could have only done it by challenging the axioms with the Matrix Columns.

If you notice, very carefully that in Old Math, there really is no difference between Successor Function and Mathematical Induction. So that one of them is rather superfluous. You can get Math Induction by just Successor, or, you can get Successor by just Math Induction, and that you do not need both as axioms. But the reason Old Math was convinced you needed both, was that, they felt you could not say -- any set that looks and feels like the Counting Numbers, could not be differentiated apart from the Counting Numbers of Successor Function, unless you had Math Induction telling you so.

Well, here I discovered something truly remarkable. That you do in fact need both Successor and Math Induction, but not for the silly reasoning of Old Math.

In these columns notice that you get both Successor and Math Induction and both are vastly different::



0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6


The Successor is seen as the adding of 1 so we move from 10 Column to 11 Column by addition, by adding one more new number. This is Addition and Successor. But what is Math Induction, if not successor?

Beautifully, Math Induction is going from 10 Column to 11 Column and then to 12 Column by Multiplication. So Successor is addition but Math Induction is multiplication, for look at the Even Columns are perfect-squares of 1^2, then 2^2, then 3^2, etc etc. Math Induction is the multiplication increase by one more.

So, a axiom system of the Counting Numbers, requires a Successor of addition, but also a Successor of Multiplication which we call Mathematical Induction.

AP
b***@gmail.com
2017-08-12 18:30:55 UTC
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Nope, you dont get induction by successor. Induction is a
separate extra statement, with a different content, than

only saying there is a successor function. Induction statement
reads as follows:

P(0) & forall x(P(x) -> P(x+1)) -> forall y P(y)

You can read this:

if P is successor inductive then P is all the naturals numbers

Without the induction statement, you might have successor
inductive sets which are not the natural numbers. For example
in set theory, you might have:

succ(x) = x u {x}

So a predicate with the property:

P({}) & forall x(P(x) -> P(succ(x)))

does not imply that P is all the natural numbers. It could
be also some other set that satisfies the above property.
Post by Archimedes Plutonium
Post by Archimedes Plutonium
Now in the proof of Goldbach we recognize that the lined up Bertrand primes in each column such as 3+5 with 3*5 in the 8 column are complements, 8 complement to 15. So we have to ask whether the Goldbach is the only spot in mathematics where complements of a+b with a*b comes to prominence. Surely there must be other topics in math where complements serve a major role.
Alright, I made a beautiful breakthrough in mathematics of the Peano Axioms. And I could have only done it by challenging the axioms with the Matrix Columns.
If you notice, very carefully that in Old Math, there really is no difference between Successor Function and Mathematical Induction. So that one of them is rather superfluous. You can get Math Induction by just Successor, or, you can get Successor by just Math Induction, and that you do not need both as axioms. But the reason Old Math was convinced you needed both, was that, they felt you could not say -- any set that looks and feels like the Counting Numbers, could not be differentiated apart from the Counting Numbers of Successor Function, unless you had Math Induction telling you so.
Well, here I discovered something truly remarkable. That you do in fact need both Successor and Math Induction, but not for the silly reasoning of Old Math.
0  1
0  2
1  1
0  3
1  2
0  4
1  3
2  2
0  5
1  4
2  3
0  6
1  5
2  4
3  3
0  7
1  6
2  5
3  4
0  8
1  7
2  6
3  5
4  4
0  9
1  8
2  7
3  6
4  5
0  10
1   9
2   8
3   7
4   6
5   5
0   11
1   10
2    9
3    8
4    7
5    6
0    12
1    11
2    10
3      9
4      8
5      7
6      6
The Successor is seen as the adding of 1 so we move from 10 Column to 11 Column by addition, by adding one more new number. This is Addition and Successor. But what is Math Induction, if not successor?
Beautifully, Math Induction is going from 10 Column to 11 Column and then to 12 Column by Multiplication. So Successor is addition but Math Induction is multiplication, for look at the Even Columns are perfect-squares of 1^2, then 2^2, then 3^2, etc etc. Math Induction is the multiplication increase by one more.
So, a axiom system of the Counting Numbers, requires a Successor of addition, but also a Successor of Multiplication which we call Mathematical Induction.
AP
b***@gmail.com
2017-08-12 18:45:39 UTC
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See Judith Roitman, page 40, definition 31
Introduction to Modern Set Theory.
New York: Wiley, 1990.
Post by b***@gmail.com
P({}) & forall x(P(x) -> P(succ(x)))
does not imply that P is all the natural numbers. It could
be also some other set that satisfies the above property.
Archimedes Plutonium
2017-08-13 05:44:36 UTC
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Alright, did some looking, with no success. Looking for another spot in mathematics, where the Complements of addition versus multiplication come to center stage of attention. In proof of Goldbach the complements come center stage for if 3+5 does not exist, then 3*5 does not exist, if 7+5 does not exist, then 7*5 does not exist and that is basically how Goldbach is proven true. If there ever was a Even Number that has no two primes to add up to the even number N, means there exists two primes p and q, both of them less than or equal to .5N, and such that p*q does not exist.

So we have heavy use of Complementarity of addition versus multiplication, can it be that only in Goldbach is this feature of Complementarity encountered?

It is somewhat mystical how these complements relate to one another

3+5 = 8 , relate to 3*5 = 15

or

5+7 = 12, relate to 5*7 = 35

Kind of bizarre that 8 relates to 15, or that 12 relates to 35.

Maybe in Matrices theory, this sort of complementarity pops up all the time.

AP
Archimedes Plutonium
2017-08-13 07:45:25 UTC
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Alright, something obvious to only a few, asking where else in math is 3+5 =8 related to 3*5 =15. For it is half perimeter related to area rectangle. Maybe it was hidden because we often do not concern ourselves with a half perimeter.

AP
Archimedes Plutonium
2017-08-13 09:03:10 UTC
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Post by Archimedes Plutonium
Alright, something obvious to only a few, asking where else in math is 3+5 =8 related to 3*5 =15. For it is half perimeter related to area rectangle. Maybe it was hidden because we often do not concern ourselves with a half perimeter.
Now, where in all of science, physics, does 1/2 perimeter become essential? Not sure, but I suspect kinetic energy as 1/2mvv is the 1/2 meaning of half perimeter.

AP
Archimedes Plutonium
2017-08-13 17:45:31 UTC
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Post by Archimedes Plutonium
Post by Archimedes Plutonium
Alright, something obvious to only a few, asking where else in math is 3+5 =8 related to 3*5 =15. For it is half perimeter related to area rectangle. Maybe it was hidden because we often do not concern ourselves with a half perimeter.
Now, where in all of science, physics, does 1/2 perimeter become essential? Not sure, but I suspect kinetic energy as 1/2mvv is the 1/2 meaning of half perimeter.
Alright, here I am asking if math is soaked through, deep soaking of the idea of addition complementary to multiplication. Does that arise just in rare cases? Or is math full of that complementary. For it seems to us, observers, that 15 is not connected to 8 in any sort of profound manner. That 8 is no more important to 15, than is 79 or 222. So what gives here. Is math saturated full of where a+b is related intricately with a*b or is it just on rare occasions like Goldbach proof.

Okay, well, I found the answer. And it is vastly surprising. That 8 is especially connected to 15, or another example 10 is especially connected to 21 in 3+7 is to 3*7. This Complementarity is saturated full in Calculus. In Calculus, we are dealing with, primarily dy and dx, and dealing with dy +dx which is 1/2 a rectangle perimeter whilst the dy*dx is the well known integral of Calculus.

It is that we seldom if ever focus and reflect upon dy+dx, and no-one until now recognizes or sees the dy+dx as the 1/2 perimeter of the rectangle used to produce the integral of calculus.

So when I asked does 3+5 =8 and 3*5=15, what is the relationship of 8 to 15 have any math significance, well, all of calculus is thoroughly dependent on that Complementarity of 8 versus 15. For the 3 is one side of the rectangle, the 5, the other side for a 1/2 perimeter and the Calculus integral is the area 3*5= 15.

In my last edition of Correcting Math, I reached a chapter on dy+dx, but I never expounded on that concept, as well, and as in depth, as I have just now expounded.

AP
b***@gmail.com
2017-08-14 21:59:55 UTC
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You don't need multiplication, you only need squaring.
There is a very simple identity, which makes you go

with squaring only and no multiplication. Just if
you have two numbers x,y you can compute:

the arithmetic mean of x,y: m=(x+y)/2

the arithmetic mean of x,-y: n=(x-y)/2

We then have:

x*y = m^2 - n^2

See first page here:

A treatise on the analytical geometry
of the point, line, circle and conic sections
Casey, John (1820-1891)
http://gallica.bnf.fr/ark:/12148/bpt6k99630k/f36.item
Post by Archimedes Plutonium
Post by Archimedes Plutonium
Post by Archimedes Plutonium
Alright, something obvious to only a few, asking where else in math is 3+5 =8 related to 3*5 =15. For it is half perimeter related to area rectangle. Maybe it was hidden because we often do not concern ourselves with a half perimeter.
Now, where in all of science, physics, does 1/2 perimeter become essential? Not sure, but I suspect kinetic energy as 1/2mvv is the 1/2 meaning of half perimeter.
Alright, here I am asking if math is soaked through, deep soaking of the idea of addition complementary to multiplication. Does that arise just in rare cases? Or is math full of that complementary. For it seems to us, observers, that 15 is not connected to 8 in any sort of profound manner. That 8 is no more important to 15, than is 79 or 222. So what gives here. Is math saturated full of where a+b is related intricately with a*b or is it just on rare occasions like Goldbach proof.
Okay, well, I found the answer. And it is vastly surprising. That 8 is especially connected to 15, or another example 10 is especially connected to 21 in 3+7 is to 3*7. This Complementarity is saturated full in Calculus. In Calculus, we are dealing with, primarily dy and dx, and dealing with dy +dx which is 1/2 a rectangle perimeter whilst the dy*dx is the well known integral of Calculus.
It is that we seldom if ever focus and reflect upon dy+dx, and no-one until now recognizes or sees the dy+dx as the 1/2 perimeter of the rectangle used to produce the integral of calculus.
So when I asked does 3+5 =8 and 3*5=15, what is the relationship of 8 to 15 have any math significance, well, all of calculus is thoroughly dependent on that Complementarity of 8 versus 15. For the 3 is one side of the rectangle, the 5, the other side for a 1/2 perimeter and the Calculus integral is the area 3*5= 15.
In my last edition of Correcting Math, I reached a chapter on dy+dx, but I never expounded on that concept, as well, and as in depth, as I have just now expounded.
AP
Don Redmond
2017-08-13 14:12:58 UTC
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Post by Archimedes Plutonium
Alright, did some looking, with no success. Looking for another spot in mathematics, where the Complements of addition versus multiplication come to center stage of attention. In proof of Goldbach the complements come center stage for if 3+5 does not exist, then 3*5 does not exist, if 7+5 does not exist, then 7*5 does not exist and that is basically how Goldbach is proven true. If there ever was a Even Number that has no two primes to add up to the even number N, means there exists two primes p and q, both of them less than or equal to .5N, and such that p*q does not exist.
So we have heavy use of Complementarity of addition versus multiplication, can it be that only in Goldbach is this feature of Complementarity encountered?
It is somewhat mystical how these complements relate to one another
3+5 = 8 , relate to 3*5 = 15
or
5+7 = 12, relate to 5*7 = 35
Kind of bizarre that 8 relates to 15, or that 12 relates to 35.
Maybe in Matrices theory, this sort of complementarity pops up all the time.
AP
If you're in a ring, then of course both ab and a + b are defined so if one is defined so is the other.

Also, from your two examples it's hardly magic. You have x + (x + 2) = 2(x + 1) and x(x + 2) = (x + 1)^2. Doesn't seem to have anything to do with primes.

Don
b***@gmail.com
2017-08-13 14:59:05 UTC
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Hi,

In any first order structure with signature <D,+> where + is of
arity two, the function + is defined. It just says that we have
a function symbol +, which we will interpret as:

+: D x D -> D

Further in a structure of signature <D,+,*> where + is of arity
two and * is of arity two, both the functions + and * are
defined. It just says that we have a function symbol + and a
function symbol *, which we will interpret as:

+: D x D -> D

*: D x D -> D

The notions ring, group, field, etc.. assumes a structure, and then
they don't indicates whether +, *, etc.. is defined, this is already
done by refering to a structure. The notions ring, group, field, etc..
indicate what axioms these function symbols should satisfy.

For example an abelian group is a structure <G,°,e,^(-1)> such that:

a ° (b ° c) = (a ° b) ° c

a ° b = b ° a

a ° e = a

a ° a^(-1) = e

Refering to <G,°> as a structure, already implies that we assume
the existence of a function °. By set theory such function will
exist. But it might be that our axioms are inconsistent,

and thus that although such functions do exist, none of them satisfy
the group axioms. But it is easy to show that there is at least one
such function, take the binary group:

° | 0 1 ^(-1)
-------- ------
0 | 0 1 0 | 1
1 | 1 0 1 | 0

You can easily verify the group axioms. So we know there is at least
one set G and one function ° : G x G -> G that satisfy the group
axioms, hence the group axioms are not inconsistent.

Bye
Post by Don Redmond
Post by Archimedes Plutonium
Alright, did some looking, with no success. Looking for another spot in mathematics, where the Complements of addition versus multiplication come to center stage of attention. In proof of Goldbach the complements come center stage for if 3+5 does not exist, then 3*5 does not exist, if 7+5 does not exist, then 7*5 does not exist and that is basically how Goldbach is proven true. If there ever was a Even Number that has no two primes to add up to the even number N, means there exists two primes p and q, both of them less than or equal to .5N, and such that p*q does not exist.
So we have heavy use of Complementarity of addition versus multiplication, can it be that only in Goldbach is this feature of Complementarity encountered?
It is somewhat mystical how these complements relate to one another
3+5 = 8 , relate to 3*5 = 15
or
5+7 = 12, relate to 5*7 = 35
Kind of bizarre that 8 relates to 15, or that 12 relates to 35.
Maybe in Matrices theory, this sort of complementarity pops up all the time.
AP
If you're in a ring, then of course both ab and a + b are defined so if one is defined so is the other.
Also, from your two examples it's hardly magic. You have x + (x + 2) = 2(x + 1) and x(x + 2) = (x + 1)^2. Doesn't seem to have anything to do with primes.
Don
Archimedes Plutonium
2017-08-13 17:31:31 UTC
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Post by Don Redmond
Post by Archimedes Plutonium
Alright, did some looking, with no success. Looking for another spot in mathematics, where the Complements of addition versus multiplication come to center stage of attention. In proof of Goldbach the complements come center stage for if 3+5 does not exist, then 3*5 does not exist, if 7+5 does not exist, then 7*5 does not exist and that is basically how Goldbach is proven true. If there ever was a Even Number that has no two primes to add up to the even number N, means there exists two primes p and q, both of them less than or equal to .5N, and such that p*q does not exist.
So we have heavy use of Complementarity of addition versus multiplication, can it be that only in Goldbach is this feature of Complementarity encountered?
It is somewhat mystical how these complements relate to one another
3+5 = 8 , relate to 3*5 = 15
or
5+7 = 12, relate to 5*7 = 35
Kind of bizarre that 8 relates to 15, or that 12 relates to 35.
Maybe in Matrices theory, this sort of complementarity pops up all the time.
AP
If you're in a ring, then of course both ab and a + b are defined so if one is defined so is the other.
Also, from your two examples it's hardly magic. You have x + (x + 2) = 2(x + 1) and x(x + 2) = (x + 1)^2. Doesn't seem to have anything to do with primes.
Don
Hi Don, I figured out where you failed on the conic experiment, rolling up a sheet of paper into a cone, taping it secure, then cutting at diagonal to fetch a oval, never ellipse. You cut parallel to the base retrieving a circle.

Try it again, this time cut diagonal.

AP
Don Redmond
2017-08-14 17:48:31 UTC
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Post by Archimedes Plutonium
Post by Don Redmond
Post by Archimedes Plutonium
Alright, did some looking, with no success. Looking for another spot in mathematics, where the Complements of addition versus multiplication come to center stage of attention. In proof of Goldbach the complements come center stage for if 3+5 does not exist, then 3*5 does not exist, if 7+5 does not exist, then 7*5 does not exist and that is basically how Goldbach is proven true. If there ever was a Even Number that has no two primes to add up to the even number N, means there exists two primes p and q, both of them less than or equal to .5N, and such that p*q does not exist.
So we have heavy use of Complementarity of addition versus multiplication, can it be that only in Goldbach is this feature of Complementarity encountered?
It is somewhat mystical how these complements relate to one another
3+5 = 8 , relate to 3*5 = 15
or
5+7 = 12, relate to 5*7 = 35
Kind of bizarre that 8 relates to 15, or that 12 relates to 35.
Maybe in Matrices theory, this sort of complementarity pops up all the time.
AP
If you're in a ring, then of course both ab and a + b are defined so if one is defined so is the other.
Also, from your two examples it's hardly magic. You have x + (x + 2) = 2(x + 1) and x(x + 2) = (x + 1)^2. Doesn't seem to have anything to do with primes.
Don
Hi Don, I figured out where you failed on the conic experiment, rolling up a sheet of paper into a cone, taping it secure, then cutting at diagonal to fetch a oval, never ellipse. You cut parallel to the base retrieving a circle.
Try it again, this time cut diagonal.
AP
I did cut on the oblique and got the results reported.

Don
Archimedes Plutonium
2017-08-14 23:19:49 UTC
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Post by Don Redmond
I did cut on the oblique and got the results reported.
Don
And what results did you get, a oval or ellipse
b***@gmail.com
2017-08-16 00:46:45 UTC
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The shorter the posts of AP brain farto, the
more delusional he is. From walls of impressive

internet artefacts, to miserable incomprehensive
snippets, in only 30 years. What is the common

thing all these 30 years? Not a single line of math.
Post by Archimedes Plutonium
Post by Don Redmond
I did cut on the oblique and got the results reported.
Don
And what results did you get, a oval or ellipse
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