A243702 -id:A243702 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 17 2014

nonn

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

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)

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

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)

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

A372518 Positive numbers primitively represented by the indefinite quadratic form x^2 + 13xy - 9y^2.

Original entry on oeis.org

1, 5, 21, 39, 51, 59, 81, 91, 105, 119, 131, 139, 141, 159, 189, 195, 201, 221, 241, 255, 269, 271, 279, 291, 295, 329, 351, 359, 369, 371, 405, 409, 411, 441, 455, 459, 469, 471, 501, 541, 549, 569, 579, 595, 599, 611, 651, 655, 661, 679, 681, 689, 695, 699

Offset: 1

Peter Luschny, May 04 2024

nonn

Discriminant 205.

Peter Luschny, Binary Quadratic Forms, GitHub 2024.

Cf. A243701 (primes), A243702 (all), this sequence (primitively).

load('https://raw.githubusercontent.com/PeterLuschny/BinaryQuadraticForms/main/BinaryQF.sage')
Q = binaryQF([1, 13, -9])
print(Q.represented_positives(700, 'primitively'))

cp's OEIS Frontend

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

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