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A243701 Primes represented by the indefinite quadratic form x^2 + 13xy - 9y^2. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
Peter Luschny, Binary Quadratic Forms, GitHub 2024.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
PROG
(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
CROSSREFS
This sequence (primes), A243702 (all), A372518 (primitively).
Sequence in context: A015994 A222563 A141951 * A179028 A179029 A049079
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 17 2014
STATUS
approved

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Last modified July 23 16:47 EDT 2024. Contains 374552 sequences. (Running on oeis4.)