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A070179 Primes p such that x^2 = 2 has a solution mod p, but x^(2^2) = 2 has no solution mod p. 17
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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A139879 A281792 A293394 * A155072 A243451 A145991

Adjacent sequences:  A070176 A070177 A070178 * A070180 A070181 A070182

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 29 2002

STATUS

approved

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Last modified January 22 08:47 EST 2018. Contains 298042 sequences.