A243701 - OEIS

%I #21 May 04 2024 10:29:54

%S 5,59,131,139,241,269,271,359,409,541,569,599,661,701,761,859,881,911,

%T 941,1021,1091,1109,1291,1399,1439,1481,1549,1559,1579,1609,1619,1931,

%U 1999,2011,2029,2089,2099,2111,2141,2251,2399,2459,2521,2711,2729,2731,2749

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

%C Discriminant 205.

%C 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.

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

%H Will Jagy et al.,<a href="http://mathoverflow.net/questions/171807/positive-primes-represented-by-indefinite-binary-quadratic-form">Positive primes represented by indefinite binary quadratic form"</a>, MathOverflow # 171807, 2014.

%H Will Jagy et al., <a href="http://mathoverflow.net/questions/171846/positive-primes-represented-by-an-indefinite-binary-form-reducing-poly-degree-f">Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4</a>, MathOverflow # 171846, 2014.

%H Peter Luschny, <a href="https://github.com/PeterLuschny/BinaryQuadraticForms">Binary Quadratic Forms</a>, GitHub 2024.

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%o (PARI)

%o fc(a,b,c,M) = {

%o   my(t1=List(),t2);

%o   forprime(p=2,prime(M),

%o     t2 = qfbsolve(Qfb(a,b,c),p);

%o     if(t2 != 0, listput(t1,p))

%o   );

%o   Vec(t1)

%o };

%o fc(1,13,-9,600)

%o (SageMath)

%o load('https://raw.githubusercontent.com/PeterLuschny/BinaryQuadraticForms/main/BinaryQF.sage')

%o Q = binaryQF([1, 13, -9])

%o print(Q.represented_positives(2750, 'prime'))  # _Peter Luschny_, May 04 2024

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jun 17 2014