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A038883
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Primes p such that 13 is a square mod p.
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43
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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
| Borevich and Shafaewich, Number Theory.
D. B. Zagier, Zetafunktionen und quadratische Koerper.
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EXAMPLE
| 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
| For[a = 1, a < 1001, a++, p = Prime[a]; t = Mod[p, 13]; If[Or[t == 1, t == 3, t == 4, t == 9, t == 10, t == 12], Print[p]]] - Neil Fernandez, Jun 22 2006
Select[ Prime@ Range@ 118, JacobiSymbol[ #, 13] > -1 &] (* Robert G. Wilson v, May 16 2008 *)
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PROG
| (PARI) forprime(p=2, 1e3, if(issquare(Mod(13, p)), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011
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CROSSREFS
| Cf. A038883 (Primes p such that 13 is a square mod p) A141111, A141112 (d=65).
Sequence in context: A038956 A040123 * A141188 A019347 A184777 A045433
Adjacent sequences: A038880 A038881 A038882 * A038884 A038885 A038886
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 27 2008, Jul 28 2008
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