Post by MKKHI Guys,
I want to know other primitive roots of prime number 181 apart from 2.
Can anyone give me website link that has information about primitive
roots of prime number other than least primitive root.
Using GP/Pari:
Take your pick which method -
1) Small hammer - generate all powers of a known primitive root.
(19:53) gp > print(factor(180))
[2, 2; 3, 2; 5, 1]
(19:53) gp > znprimroot(181)
Mod(2, 181)
(19:53) gp > forstep(j=1,180,6,if(j%5!=0,print1(" ",lift(Mod(2,181)^j)));if(j%5!=1,print1(" ",lift(Mod(2,181)^(j+4)))))
2 128 57 47 28 112 163 115 98 118 78 131 105 58 23 24 96 171 21 84 77 127 41 90 179 53 124 134 153 69 18 66 83 63 103 50 76 123 158 157 85 10 160 97 104 54 140 91
2) Big hammer 1 - factor the 180th cyclotomic polynomial over Z181.
(19:57) gp > lift(factormod(polcyclo(180),181))[,1]
[x + 2, x + 10, x + 18, x + 21, x + 23, x + 24, x + 28, x + 41, x + 47, x + 50, x + 53, x + 54, x + 57, x + 58, x + 63, x + 66, x + 69, x + 76, x + 77, x + 78, x + 83, x + 84, x + 85, x + 90, x + 91, x + 96, x + 97, x + 98, x + 103, x + 104, x + 105, x + 112, x + 115, x + 118, x + 123, x + 124, x + 127, x + 128, x + 131, x + 134, x + 140, x + 153, x + 157, x + 158, x + 160, x + 163, x + 171, x + 179]~
3) Big hammer 2 - find all the roots of the 180th cyclotomic polynomial mod 181.
(19:59) gp > lift(polrootsmod(polcyclo(180),181))
[2, 10, 18, 21, 23, 24, 28, 41, 47, 50, 53, 54, 57, 58, 63, 66, 69, 76, 77, 78, 83, 84, 85, 90, 91, 96, 97, 98, 103, 104, 105, 112, 115, 118, 123, 124, 127, 128, 131, 134, 140, 153, 157, 158, 160, 163, 171, 179]~
4) Dumb brute force.
(20:03) gp > for(i=1,180,if(znorder(Mod(i,181))==180,print1(" "i)))
2 10 18 21 23 24 28 41 47 50 53 54 57 58 63 66 69 76 77 78 83 84 85 90 91 96 97 98 103 104 105 112 115 118 123 124 127 128 131 134 140 153 157 158 160 163 171 179
There are probably other ways.
Phil
--
They no longer do my traditional winks tournament lunch - liver and bacon.
It's just what you need during a winks tournament lunchtime to replace lost
... liver. -- Anthony Horton, 2004/08/27 at the Cambridge 'Long Vac.'