

A163186


Primes p such that the equation x^64 == 2 (mod p) has a solution, and ord_p(2) is even.


2



25601, 50177, 59393, 65537, 96001, 115201, 195457, 262657, 266369, 267649, 279553, 286721, 295937, 299393, 306689, 331777, 366593, 425857, 460289, 495617, 509569, 525313, 528001, 556289, 595073, 710273, 730753, 790529, 819457, 868993, 1021697, 1022977, 1049089
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OFFSET

1,1


COMMENTS

Such primes are the exceptional p for which x^64 == 2 (mod p) has a solution, as x^64 == 2 (mod p) is soluble for *every* p with ord_p(2) odd.
But if ord_p(2) is even and p  1 = 2^r.j with j odd, then x^64 == 2 (mod p) is soluble if and only if ord_p(2) is not divisible by 2^(r5). See comment at A163185 for explanation.
Most primes p for which x^64 == 2 (mod p) has a solution (A051101) have ord_p(2) odd (so belong to A163183). Thus 25601 (first element of current sequence, and 827th element of A051101) is the first element where A051101 and A163183 differ.


LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..1000


EXAMPLE

For p = 25601, 562^64 == 2 (mod p), the 2power part of p1 is 2^10 and ord_p(2) = 400, which is even but has 2power part 2^4, which is not divisible by 2^(105).


MAPLE

with(numtheory):k:=6: A:=NULL:p:=2: for c to 30000 do p:=nextprime(p); o:=order(2, p); R:=gcd(2^100, p1); if o mod 2=0 and p mod 2^(k+1) = 1 and o mod R/2^(k1)<>0 then A:=A, p; fi; od:A;


PROG

(PARI) lista(nn) = forprime(p=3, nn, if(znorder(Mod(2, p))%2==0 && []~!=polrootsmod(x^64+2, p), print1(p, ", "))); \\ Jinyuan Wang, Mar 24 2020


CROSSREFS

A051101 (all primes p for which x^62 == 2 (mod p) has a solution) is a union of A163183 (primes p with ord_p(2) odd) and the current sequence.
Sequence in context: A210093 A184027 A023350 * A232838 A180299 A172644
Adjacent sequences: A163183 A163184 A163185 * A163187 A163188 A163189


KEYWORD

nonn,easy


AUTHOR

Christopher J. Smyth, Jul 24 2009


EXTENSIONS

More terms from Jinyuan Wang, Mar 24 2020


STATUS

approved



