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A097955
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Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).
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4
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3, 13, 31, 37, 41, 43, 53, 67, 71, 79, 83, 89, 107, 151, 157, 163, 173, 191, 197, 199, 227, 239, 241, 271, 277, 281, 283, 293, 307, 311, 317, 347, 359, 373, 397, 401, 409, 431, 439, 443, 449, 467, 479, 521, 523, 547, 557, 563, 569, 587, 599, 601, 613, 631, 641
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OFFSET
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1,1
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COMMENTS
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Also 3 and primes p such that (p^2 - 1)/24 mod 10 = {0, 7}. - Richard R. Forberg, Aug 31 2013
Also primes p such that x^2 = 10 mod p has integer solutions, or Legendre(10, p) = 1. However, p could be irreducible but not prime in Z[sqrt(10)], especially if p = 3 or 7 mod 10. - Alonso del Arte, Dec 27 2015
Rational primes that decompose in the field Q(sqrt(10)). - N. J. A. Sloane, Dec 26 2017
Primes p such that kronecker(10,p) = 1 (or equivalently, kronecker(40,p) = 1).
Primes congruent to 1, 3, 9, 13, 27, 31, 37, 39 modulo 40. (End)
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LINKS
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EXAMPLE
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For p = 13, 5^6 - 2^6 = 15561 is divisible by 13, so 13 is in the sequence.
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MAPLE
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select(p -> isprime(p) and 10 &^ ((p-1)/2) mod p = 1, [seq(i, i=3..1000, 2)]); # Robert Israel, Dec 28 2015
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MATHEMATICA
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Select[Prime[Range[100]], JacobiSymbol[10, #] == 1 &] (* Alonso del Arte, Dec 27 2015 *)
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PROG
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(PARI) \\ s = +-1, d=diff
ptopm1d2(n, x, d, s) = { forprime(p=3, n, p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0, print1(p", "))) }
ptopm1d2(1000, 5, 3, -1)
(PARI) isA097955(p) == isprime(p) && kronecker(10, p) == 1 \\ Jianing Song, Oct 13 2022
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CROSSREFS
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A038879, the sequence of primes that do not remain inert in the field Q(sqrt(10)), is essentially the same.
Cf. A038880 (rational primes that remain inert in the field Q(sqrt(10))).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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