A033213 Primes of form x^2+17*y^2.

17, 53, 149, 157, 281, 293, 349, 353, 409, 461, 509, 569, 593, 613, 661, 733, 797, 829, 937, 977, 1097, 1237, 1361, 1381, 1409, 1453, 1597, 1709, 1721, 1733, 1753, 1777, 1861, 2053, 2089, 2129, 2141

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from the first term, odd primes p such that (-17/p) = 1 and x^4 - x^2 = 4 has a solution mod p. - Charles R Greathouse IV, Nov 11 2012

All terms are in {1, 9, 13, 17, 21, 25, 33, 49, 53} mod 68, but this is not sufficient for inclusion. - Charles R Greathouse IV, Nov 11 2012

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Programs

Magma

/* By first comment: */ [17] cat [p: p in PrimesInInterval(3,2200) | LegendreSymbol(-17, p) eq 1 and exists{x: x in ResidueClassRing(p) | x^4-x^2 eq 4}]; // Bruno Berselli, Nov 11 2012

Mathematica

QuadPrimes2[1, 0, 17, 10000] (* see A106856 *)

PARI

is(n)=if(kronecker(17,n)>0 && kronecker(-17,n)>0 && n>2 && isprime(n), kronecker(lift((1+sqrt(Mod(17,n)))/2),n)>0, n==17) \\ Charles R Greathouse IV, Nov 11 2012