A243701 Primes represented by the indefinite quadratic form x^2 + 13xy - 9y^2.

5, 59, 131, 139, 241, 269, 271, 359, 409, 541, 569, 599, 661, 701, 761, 859, 881, 911, 941, 1021, 1091, 1109, 1291, 1399, 1439, 1481, 1549, 1559, 1579, 1609, 1619, 1931, 1999, 2011, 2029, 2089, 2099, 2111, 2141, 2251, 2399, 2459, 2521, 2711, 2729, 2731, 2749

Offset: 1

Author

N. J. A. Sloane, Jun 17 2014

Keywords

• nonn

Comments

Discriminant 205.

Comment from Noam D. Elkies, Jun 14 2014 (See the MathOverflow #171807 link): These are exactly the primes p such that the polynomial x^8+15x^6+48x^4+15x^2+1 factors into linear factors mod p.

4*a(n) has the form z^2 - 205*y^2, where z = 2*x+13*y. - Bruno Berselli, Jun 20 2014

Links

• Will Jagy et al.,Positive primes represented by indefinite binary quadratic form", MathOverflow # 171807, 2014.
• Will Jagy et al., Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4, MathOverflow # 171846, 2014.
• Peter Luschny, Binary Quadratic Forms, GitHub 2024.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

This sequence (primes), A243702 (all), A372518 (primitively).

Programs

PARI

fc(a,b,c,M) = {
  my(t1=List(),t2);
  forprime(p=2,prime(M),
    t2 = qfbsolve(Qfb(a,b,c),p);
    if(t2 != 0, listput(t1,p))
  );
  Vec(t1)
};
fc(1,13,-9,600)

SageMath

load('https://raw.githubusercontent.com/PeterLuschny/BinaryQuadraticForms/main/BinaryQF.sage')
Q = binaryQF([1, 13, -9])
print(Q.represented_positives(2750, 'prime'))  # Peter Luschny, May 04 2024