A070179 Primes p such that x^2 = 2 has a solution mod p, but x^(2^2) = 2 has no solution mod p.

17, 41, 97, 137, 193, 241, 313, 401, 409, 433, 449, 457, 521, 569, 641, 673, 761, 769, 809, 857, 929, 953, 977, 1009, 1129, 1297, 1321, 1361, 1409, 1489, 1657, 1697, 1873, 1993, 2017, 2081, 2137, 2153, 2161, 2297, 2377, 2417, 2521, 2609, 2617, 2633, 2713

Offset: 1

Author

Klaus Brockhaus, Apr 29 2002

Keywords

• nonn
• easy

Comments

Complement of A014754 with regard to primes of the form 8*k+1.

These appear to be the primes p for which 4^((p-1)*n/8) mod p = (p-2)*( n mod 2)+1. For example, 4^(5*n) mod 41 = 1,40,1,40,1,40...= 39*(n mod 2)+1 and 4^(30*n) mod 241 = 1,240,1,240,1,240...= 239*(n mod 2) +1. - Gary Detlefs, Jul 06 2014

Primes p == 1 mod 8 such that 2^((p-1)/4) == -1 mod p. - Robert Israel, Jul 06 2014

A very similar sequence is A293394. - Jonas Kaiser, Nov 08 2017

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A038873, A040098, A040100, A059667, A070180-A070188, A014754.

Programs

Magma

[p: p in PrimesUpTo(3000) | not exists{x: x in ResidueClassRing(p) | x^4 eq 2} and exists{x: x in ResidueClassRing(p) | x^2 eq 2}]; // Vincenzo Librandi, Sep 21 2012

Maple

select(p -> isprime(p) and 2 &^((p-1)/4) mod p = p-1, [8*k+1$k=1..10000]); # Robert Israel, Jul 06 2014

PARI

forprime(p=2,2720,x=0; while(x
				

PARI

{a(n) = local(m, c, x); if( n<1, 0, c = 0; m = 1; while( cMichael Somos, Mar 22 2008 */

PARI

ok(p, r, k1, k2)={
if ( Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
if ( Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
return(1);
}
forprime(p=2,10^4, if (ok(p,2,2,2^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

PARI

is(n)=n%8==1 && Mod(2,n)^(n\4)==-1 && isprime(n) \\ Charles R Greathouse IV, Nov 10 2017

Formula

Primes of the form 8*k + 1 but not x^2 + 64*y^2. - Michael Somos, Mar 22 2008

a(n) ~ 8n log n. - Charles R Greathouse IV, Nov 10 2017