A014754 Primes p == 1 mod 8 such that 2 and -2 are both 4th powers (one implies other) mod p.

73, 89, 113, 233, 257, 281, 337, 353, 577, 593, 601, 617, 881, 937, 1033, 1049, 1097, 1153, 1193, 1201, 1217, 1249, 1289, 1433, 1481, 1553, 1601, 1609, 1721, 1753, 1777, 1801, 1889, 1913, 2089, 2113, 2129, 2273, 2281, 2393, 2441, 2473, 2593, 2657, 2689

Offset: 1

Author

Warren D. Smith

Keywords

• nonn

Comments

Primes p such that x^4 == 2 has more than two (in fact four) solutions mod p. This is the sequence of terms common to A040098 (primes p such that x^4 == 2 has a solution mod p) and A007519 (primes of form 8n+1). Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 == 2 iff p - i is a solution mod p of x^4 == 2, thus the sum of first and fourth solution is p and so is the sum of second and third solution. The solutions are given in A065909, A065910, A065911 and A065912. - Klaus Brockhaus, Nov 28 2001

Primes of the form x^2+64y^2. - T. D. Noe, May 13 2005

Links

• N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..9769 (the first 1000 terms were found by Vincenzo Librandi)

Crossrefs

Cf. A040098, A007519, A014754, A007522, A065909, A065910, A065911, A065912, A070179.

Programs

PARI

A014754(m) = local(p,s,x,z); forprime(p = 3,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); z = matsize(s)[2]; if(z>2,print1(p,",")))

PARI

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

PARI

forprime(p=1, 9999, p%8==1&&ispower(Mod(2, p), 4)&&print1(p", ")) \\ M. F. Hasler, Feb 18 2014

PARI

is_A014754(p)={p%8==1&&ispower(Mod(2, p), 4)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

Extensions

Removed erroneous Mma program; extended b-file using first PARI program of M. F. Hasler. - N. J. A. Sloane, Jun 06 2014