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A243701
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Primes represented by the indefinite quadratic form x^2 + 13xy - 9y^2.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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