OFFSET
1,1
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
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
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
PROG
(PARI) forprime(p=2, 2720, x=0; while(x<p&&x^2%p!=2%p, x++); if(x<p, y=0; while(y<p&&y^(2^2)%p!=2%p, y++); if(y==p, print1(p, ", "))))
(PARI) {a(n) = local(m, c, x); if( n<1, 0, c = 0; m = 1; while( c<n, m++; if( isprime(m) && m%8 == 1, x = 0; for(y=1, sqrtint( m \ 64 ), if( issquare( m - 64 * y^2, &x), break)); if( !x, c++ ))); m)} /* Michael 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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 29 2002
STATUS
approved