The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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^(r-5). 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 2-power part of p-1 is 2^10 and ord_p(-2) = 400, which is even but has 2-power part 2^4, which is not divisible by 2^(10-5). 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, p-1); if o mod 2=0 and p mod 2^(k+1) = 1 and o mod R/2^(k-1)<>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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 1 19:48 EDT 2022. Contains 357172 sequences. (Running on oeis4.)