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A038883
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Odd primes p such that 13 is a square mod p.
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46
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3, 13, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641
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OFFSET
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1,1
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COMMENTS
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Equivalently, by quadratic reciprocity (since 13 == 1 (mod 4)), primes p which are squares mod 13.
The squares mod 13 are 0, 1, 4, 9, 3, 12 and 10.
Also primes of the form x^2 + 3*x*y - y^2. Discriminant = 13. Class = 1. This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 06 2008. R. J. Mathar proved that this coincides with the present sequence, Jul 22 2008
Primes p such that x^2 + x = 3 has a solution mod p (the solutions over the reals are (-1+-sqrt(13))/2). - Joerg Arndt, Jul 27 2011
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REFERENCES
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Z. I. Borevich and I. R. Shafarevich, Number Theory.
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LINKS
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FORMULA
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EXAMPLE
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13 == 1 (mod 3) and 1 is a square, so 3 is on the list.
101 is prime and congruent to 7^2 = 49 == 10 (mod 13), so 101 is on the list.
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MATHEMATICA
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Select[ Prime@ Range@ 118, JacobiSymbol[ #, 13] > -1 &] (* Robert G. Wilson v, May 16 2008 *)
Select[Flatten[Table[13n + {1, 3, 4, 9, 10, 12}, {n, 50}]], PrimeQ[#] &] (* Alonso del Arte, Sep 16 2012 *)
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PROG
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(PARI) select( {is_A038883(n)=bittest(5659, n%13)&&isprime(n)}, [0..666]) \\ M. F. Hasler, Feb 17 2022
(Sage) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 3, -1])
print(Q.represented_positives(641, 'prime')) # Peter Luschny, Sep 20 2018
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CROSSREFS
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For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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