Archimedes Plutonium

2017-08-08 07:31:14 UTC

Alright, so, let us go back in time, a time in which no Goldbach was around to make a conjecture. A time in which only dinosaurs were around, no math professors to fill a mind with blither blather nonsense.

So here we are in fresh new math, and we have Arithmetic and we have Unique Prime Factorization.

So we ask the simple question of from 0 to 100 how many even numbers are there and there are 50, but we pare away the 2, 4, because we want only oven numbers that are the sum of two primes, so that leaves us with 48 even numbers from 6 to 100.

Now we start the program and we abbreviate unique prime factorization as UPFAT.

Now, we include a column of Unique Prime Addition, where we borrow one of the multiplication primes and craft a number equal to the even number, call it UPADD. Now if there is no prime in UPFAT to craft UPADD we take the next higher prime and use it. Now if the primes of UPFAT do not work we bump up one of the primes to the next higher until something does work.

Even UPFAT UPADD

6 2*3 3+3

8 2*2*2 3+5

10 2*5 5*5

12 2*2*3 5+7

14 2*7 7+7

16 2*2*2*2 3+13

18 2*3*3 5+13

20 2*2*5 7+13

22 2*11 11+11

24 2*2*2*3 5+19

Now, we know the odd primes from 3 to 100 is this list of 24 primes 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

So there are 24 primes to work with to build all the even numbers from 6 to 100.

Now we ask, what are all the Two Prime Composites from 9 to 100, deleting all 2s

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 = 28

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

There are 19 such Two Prime Composites and all the evens from 6 to 34 are there except missing 30. The 30 will be picked up by 7*23 or by 11*19.

Alright we are ready for a preliminary Theorem proof of Goldbach

Preliminary Theorem Statement:: given any even number from 6 onwards, it has to have at least two primes that sum to that even number

Preliminary Proof Statement:: Every even number is dissolvable into unique prime factors. Every two odd numbers when added produce an even number. Every even number N, starting with 6, has a prime number P smaller than itself, thus, we find a second prime number Q to add on, giving P+ Q = N, and if not, then the number P*Q is also, nonexistent, violating the Fundamental Theorem of Arithmetic.

This is the Goldbach conjecture and proof, where we find just one solution set for each even number.

Now, to Prove the Goldbach extensions and Legendre and Legendre extensions we need the Columns of Even Numbers and the Bertrand Postulate, between N and 2N exists a prime.

The Legendre Extension Conjecture that between all Perfect Squares Interval, rests a Two Prime Composite, starting with 4 to 9, a Two Prime Composite all of which have form P*Q and where the P is always 3.

What we do that is new, is define Monotone Increase as per a sequence of counts where there is no zero count in an interval such as this shown below:

9

Counter = 1

16

Counter = 1

25

Counter = 1

36

Counter = 1

49

Counter = 2

64

Counter = 1

81

Counter = 2

100

Counter = 1

121

Counter = 3

144

Counter = 1

169

Counter = 2

196

Counter = 3

225

Counter = 2

256

Counter = 1

289

Counter = 4

324

Counter = 2

361

Counter = 2

400

Counter = 2

441

Counter = 3

484

Counter = 3

529

Counter = 3

576

Counter = 3

625

Counter = 2

676

Counter = 5

729

Counter = 2

784

Counter = 4

841

Counter = 3

900

Counter = 4

961

Counter = 2

1024

Counter = 4

1089

Counter = 4

1156

Counter = 3

1225

Counter = 4

1296

Counter = 4

1369

Counter = 5

1444

Counter = 4

1521

Counter = 3

1600

Counter = 3

1681

Counter = 5

1764

Counter = 5

1849

Counter = 5

1936

Counter = 5

2025

Counter = 4

2116

Counter = 4

2209

Counter = 5

2304

Counter = 4

2401

Counter = 6

2500

Counter = 5

2601

Counter = 4

2704

Counter = 4

2809

Counter = 6

2916

Counter = 4

3025

Counter = 7

3136

Counter = 5

3249

Counter = 7

3364

Counter = 4

3481

Counter = 5

3600

Counter = 7

3721

Counter = 4

3844

Counter = 9

3969

Counter = 2

4096

Counter = 4

4225

Counter = 8

4356

Counter = 8

4489

Counter = 4

4624

Counter = 8

4761

Counter = 8

4900

Counter = 5

5041

Counter = 6

5184

Counter = 5

5329

Counter = 7

5476

Counter = 6

5625

Counter = 6

5776

Counter = 5

5929

Counter = 9

6084

Counter = 5

6241

Counter = 9

6400

Counter = 6

6561

Counter = 6

6724

Counter = 8

6889

Counter = 8

7056

Counter = 8

7225

Counter = 7

7396

Counter = 5

7569

Counter = 7

7744

Counter = 6

7921

Counter = 11

8100

Counter = 9

8281

Counter = 8

8464

Counter = 7

8649

Counter = 7

8836

Counter = 7

9025

Counter = 8

9216

Counter = 6

9409

Counter = 7

9604

Counter = 9

9801

Counter = 9

10000

Counter = 8

10201

Counter = 8

10404

Counter = 8

10609

Counter = 9

10816

Counter = 10

11025

Counter = 9

11236

Counter = 7

11449

...

4277814025

Counter = 2064

4277944836

Counter = 2077

4278075649

Counter = 2108

4278206464

Counter = 2129

4278337281

Counter = 2039

4278468100

Counter = 2025

4278598921

Counter = 2098

4278729744

Counter = 2014

4278860569

Counter = 2065

4278991396

Counter = 2056

4279122225

Counter = 2061

4279253056

Counter = 2012

4279383889

Counter = 2055

4279514724

Counter = 2112

4279645561

Counter = 2066

4279776400

Counter = 2050

4279907241

Counter = 2063

4280038084

Counter = 2014

4280168929

Counter = 2089

4280299776

Counter = 2036

4280430625

Counter = 2050

4280561476

Counter = 2000

4280692329

Counter = 2099

4280823184

Counter = 2093

4280954041

Counter = 2062

4281084900

Counter = 2084

4281215761

Counter = 2052

4281346624

Counter = 2106

4281477489

Counter = 2095

4281608356

Counter = 2073

4281739225

Counter = 2052

4281870096

Counter = 2064

4282000969

Counter = 2026

4282131844

Counter = 2080

4282262721

Counter = 2055

4282393600

Counter = 2071

4282524481

Counter = 2058

4282655364

Counter = 2043

4282786249

Counter = 2088

4282917136

Counter = 2071

4283048025

Counter = 2073

4283178916

Counter = 2069

4283309809

Counter = 2056

4283440704

Counter = 2097

4283571601

Counter = 2061

4283702500

Counter = 2120

4283833401

Counter = 2091

4283964304

Counter = 2091

4284095209

Counter = 2098

4284226116

Counter = 2023

4284357025

Counter = 2052

4284487936

Counter = 2093

4284618849

Counter = 2042

4284749764

Counter = 2074

4284880681

Counter = 2086

4285011600

Counter = 2041

4285142521

Counter = 2057

4285273444

Counter = 2088

4285404369

Counter = 2049

4285535296

Counter = 2104

4285666225

Counter = 2067

4285797156

Counter = 2061

4285928089

Counter = 2068

4286059024

Counter = 2073

4286189961

Counter = 2036

4286320900

Counter = 2077

4286451841

Counter = 2044

4286582784

Counter = 2086

4286713729

Counter = 2099

4286844676

Counter = 2108

4286975625

Counter = 2083

4287106576

Counter = 2051

4287237529

Counter = 2083

4287368484

Counter = 2060

4287499441

Counter = 2068

4287630400

Counter = 2086

4287761361

Counter = 2053

4287892324

Counter = 2105

4288023289

Counter = 2087

4288154256

Counter = 2065

4288285225

Counter = 2074

4288416196

Counter = 2020

4288547169

Counter = 2126

4288678144

Counter = 2090

4288809121

Counter = 2115

4288940100

Counter = 2090

4289071081

Counter = 2088

4289202064

Counter = 2077

4289333049

Counter = 2125

4289464036

Counter = 2008

4289595025

Counter = 2080

4289726016

Counter = 2042

4289857009

Counter = 2086

4289988004

Counter = 2088

4290119001

Counter = 2058

4290250000

Counter = 2036

4290381001

Counter = 2149

4290512004

Counter = 2067

4290643009

Counter = 2078

4290774016

Counter = 2060

4290905025

Counter = 2089

4291036036

Counter = 2058

4291167049

Counter = 2118

4291298064

Counter = 1992

4291429081

Counter = 2031

4291560100

Counter = 2078

4291691121

Counter = 2090

4291822144

Counter = 2046

4291953169

Counter = 2117

4292084196

Counter = 2091

4292215225

Counter = 2098

4292346256

Counter = 2074

4292477289

Counter = 2047

4292608324

Counter = 2087

4292739361

Counter = 2090

4292870400

Counter = 2065

4293001441

Counter = 2058

4293132484

Counter = 2086

4293263529

Counter = 2059

4293394576

Counter = 2045

4293525625

Counter = 2064

4293656676

Counter = 2058

4293787729

Counter = 2049

4293918784

Counter = 2013

4294049841

Counter = 2126

4294180900

Counter = 2125

4294311961

Counter = 2038

4294443024

Counter = 2093

4294574089

Counter = 2095

4294705156

Counter = 2089

4294836225

(source Vinicius)

AP

So here we are in fresh new math, and we have Arithmetic and we have Unique Prime Factorization.

So we ask the simple question of from 0 to 100 how many even numbers are there and there are 50, but we pare away the 2, 4, because we want only oven numbers that are the sum of two primes, so that leaves us with 48 even numbers from 6 to 100.

Now we start the program and we abbreviate unique prime factorization as UPFAT.

Now, we include a column of Unique Prime Addition, where we borrow one of the multiplication primes and craft a number equal to the even number, call it UPADD. Now if there is no prime in UPFAT to craft UPADD we take the next higher prime and use it. Now if the primes of UPFAT do not work we bump up one of the primes to the next higher until something does work.

Even UPFAT UPADD

6 2*3 3+3

8 2*2*2 3+5

10 2*5 5*5

12 2*2*3 5+7

14 2*7 7+7

16 2*2*2*2 3+13

18 2*3*3 5+13

20 2*2*5 7+13

22 2*11 11+11

24 2*2*2*3 5+19

Now, we know the odd primes from 3 to 100 is this list of 24 primes 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

So there are 24 primes to work with to build all the even numbers from 6 to 100.

Now we ask, what are all the Two Prime Composites from 9 to 100, deleting all 2s

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 = 28

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

There are 19 such Two Prime Composites and all the evens from 6 to 34 are there except missing 30. The 30 will be picked up by 7*23 or by 11*19.

Alright we are ready for a preliminary Theorem proof of Goldbach

Preliminary Theorem Statement:: given any even number from 6 onwards, it has to have at least two primes that sum to that even number

Preliminary Proof Statement:: Every even number is dissolvable into unique prime factors. Every two odd numbers when added produce an even number. Every even number N, starting with 6, has a prime number P smaller than itself, thus, we find a second prime number Q to add on, giving P+ Q = N, and if not, then the number P*Q is also, nonexistent, violating the Fundamental Theorem of Arithmetic.

This is the Goldbach conjecture and proof, where we find just one solution set for each even number.

Now, to Prove the Goldbach extensions and Legendre and Legendre extensions we need the Columns of Even Numbers and the Bertrand Postulate, between N and 2N exists a prime.

The Legendre Extension Conjecture that between all Perfect Squares Interval, rests a Two Prime Composite, starting with 4 to 9, a Two Prime Composite all of which have form P*Q and where the P is always 3.

What we do that is new, is define Monotone Increase as per a sequence of counts where there is no zero count in an interval such as this shown below:

9

Counter = 1

16

Counter = 1

25

Counter = 1

36

Counter = 1

49

Counter = 2

64

Counter = 1

81

Counter = 2

100

Counter = 1

121

Counter = 3

144

Counter = 1

169

Counter = 2

196

Counter = 3

225

Counter = 2

256

Counter = 1

289

Counter = 4

324

Counter = 2

361

Counter = 2

400

Counter = 2

441

Counter = 3

484

Counter = 3

529

Counter = 3

576

Counter = 3

625

Counter = 2

676

Counter = 5

729

Counter = 2

784

Counter = 4

841

Counter = 3

900

Counter = 4

961

Counter = 2

1024

Counter = 4

1089

Counter = 4

1156

Counter = 3

1225

Counter = 4

1296

Counter = 4

1369

Counter = 5

1444

Counter = 4

1521

Counter = 3

1600

Counter = 3

1681

Counter = 5

1764

Counter = 5

1849

Counter = 5

1936

Counter = 5

2025

Counter = 4

2116

Counter = 4

2209

Counter = 5

2304

Counter = 4

2401

Counter = 6

2500

Counter = 5

2601

Counter = 4

2704

Counter = 4

2809

Counter = 6

2916

Counter = 4

3025

Counter = 7

3136

Counter = 5

3249

Counter = 7

3364

Counter = 4

3481

Counter = 5

3600

Counter = 7

3721

Counter = 4

3844

Counter = 9

3969

Counter = 2

4096

Counter = 4

4225

Counter = 8

4356

Counter = 8

4489

Counter = 4

4624

Counter = 8

4761

Counter = 8

4900

Counter = 5

5041

Counter = 6

5184

Counter = 5

5329

Counter = 7

5476

Counter = 6

5625

Counter = 6

5776

Counter = 5

5929

Counter = 9

6084

Counter = 5

6241

Counter = 9

6400

Counter = 6

6561

Counter = 6

6724

Counter = 8

6889

Counter = 8

7056

Counter = 8

7225

Counter = 7

7396

Counter = 5

7569

Counter = 7

7744

Counter = 6

7921

Counter = 11

8100

Counter = 9

8281

Counter = 8

8464

Counter = 7

8649

Counter = 7

8836

Counter = 7

9025

Counter = 8

9216

Counter = 6

9409

Counter = 7

9604

Counter = 9

9801

Counter = 9

10000

Counter = 8

10201

Counter = 8

10404

Counter = 8

10609

Counter = 9

10816

Counter = 10

11025

Counter = 9

11236

Counter = 7

11449

...

4277814025

Counter = 2064

4277944836

Counter = 2077

4278075649

Counter = 2108

4278206464

Counter = 2129

4278337281

Counter = 2039

4278468100

Counter = 2025

4278598921

Counter = 2098

4278729744

Counter = 2014

4278860569

Counter = 2065

4278991396

Counter = 2056

4279122225

Counter = 2061

4279253056

Counter = 2012

4279383889

Counter = 2055

4279514724

Counter = 2112

4279645561

Counter = 2066

4279776400

Counter = 2050

4279907241

Counter = 2063

4280038084

Counter = 2014

4280168929

Counter = 2089

4280299776

Counter = 2036

4280430625

Counter = 2050

4280561476

Counter = 2000

4280692329

Counter = 2099

4280823184

Counter = 2093

4280954041

Counter = 2062

4281084900

Counter = 2084

4281215761

Counter = 2052

4281346624

Counter = 2106

4281477489

Counter = 2095

4281608356

Counter = 2073

4281739225

Counter = 2052

4281870096

Counter = 2064

4282000969

Counter = 2026

4282131844

Counter = 2080

4282262721

Counter = 2055

4282393600

Counter = 2071

4282524481

Counter = 2058

4282655364

Counter = 2043

4282786249

Counter = 2088

4282917136

Counter = 2071

4283048025

Counter = 2073

4283178916

Counter = 2069

4283309809

Counter = 2056

4283440704

Counter = 2097

4283571601

Counter = 2061

4283702500

Counter = 2120

4283833401

Counter = 2091

4283964304

Counter = 2091

4284095209

Counter = 2098

4284226116

Counter = 2023

4284357025

Counter = 2052

4284487936

Counter = 2093

4284618849

Counter = 2042

4284749764

Counter = 2074

4284880681

Counter = 2086

4285011600

Counter = 2041

4285142521

Counter = 2057

4285273444

Counter = 2088

4285404369

Counter = 2049

4285535296

Counter = 2104

4285666225

Counter = 2067

4285797156

Counter = 2061

4285928089

Counter = 2068

4286059024

Counter = 2073

4286189961

Counter = 2036

4286320900

Counter = 2077

4286451841

Counter = 2044

4286582784

Counter = 2086

4286713729

Counter = 2099

4286844676

Counter = 2108

4286975625

Counter = 2083

4287106576

Counter = 2051

4287237529

Counter = 2083

4287368484

Counter = 2060

4287499441

Counter = 2068

4287630400

Counter = 2086

4287761361

Counter = 2053

4287892324

Counter = 2105

4288023289

Counter = 2087

4288154256

Counter = 2065

4288285225

Counter = 2074

4288416196

Counter = 2020

4288547169

Counter = 2126

4288678144

Counter = 2090

4288809121

Counter = 2115

4288940100

Counter = 2090

4289071081

Counter = 2088

4289202064

Counter = 2077

4289333049

Counter = 2125

4289464036

Counter = 2008

4289595025

Counter = 2080

4289726016

Counter = 2042

4289857009

Counter = 2086

4289988004

Counter = 2088

4290119001

Counter = 2058

4290250000

Counter = 2036

4290381001

Counter = 2149

4290512004

Counter = 2067

4290643009

Counter = 2078

4290774016

Counter = 2060

4290905025

Counter = 2089

4291036036

Counter = 2058

4291167049

Counter = 2118

4291298064

Counter = 1992

4291429081

Counter = 2031

4291560100

Counter = 2078

4291691121

Counter = 2090

4291822144

Counter = 2046

4291953169

Counter = 2117

4292084196

Counter = 2091

4292215225

Counter = 2098

4292346256

Counter = 2074

4292477289

Counter = 2047

4292608324

Counter = 2087

4292739361

Counter = 2090

4292870400

Counter = 2065

4293001441

Counter = 2058

4293132484

Counter = 2086

4293263529

Counter = 2059

4293394576

Counter = 2045

4293525625

Counter = 2064

4293656676

Counter = 2058

4293787729

Counter = 2049

4293918784

Counter = 2013

4294049841

Counter = 2126

4294180900

Counter = 2125

4294311961

Counter = 2038

4294443024

Counter = 2093

4294574089

Counter = 2095

4294705156

Counter = 2089

4294836225

(source Vinicius)

AP