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 A003629 Primes p == +- 3 (mod 8), or, primes p such that 2 is not a square mod p. (Formerly M2472) 25
 3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 251, 269, 277, 283, 293, 307, 317, 331, 347, 349, 373, 379, 389, 397, 419, 421, 443, 461, 467, 491, 499, 509, 523, 541, 547, 557, 563 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Complement of A038873 relative to A000040. Also primes p such that p divides 2^((p-1)/2) + 1. - Cino Hilliard, Sep 04 2004 Primes p such that p^2 == 25 (mod 48), n > 1. - Gary Detlefs, Dec 29 2011 This sequence gives the primes p which satisfy C(p, x = 0) = -1, where C(p, x) is the minimal polynomial of 2*cos(Pi/p) (see A187360). For a proof see a comment on C(n, 0) in A230075. - Wolfdieter Lang, Oct 24 2013 Except for the initial 3, these are the primes p such that Fibonacci(p) mod 6 = 5. - Gary Detlefs, May 26 2014 Inert rational primes in the field Q(sqrt(2)). - N. J. A. Sloane, Dec 26 2017 If a prime p is congruent to 3 or 5 (mod 8) and r > 1, then 2^((p-1)*p^(r-1)/2) == -1 (mod p^r). - Marina Ibrishimova, Sep 29 2018 For the proofs or the comments by Cino Hilliard and Marina Ibrishimova, see link below. - Robert Israel, Apr 24 2019 REFERENCES Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870. Ronald S. Irving, Integers, Polynomials, and Rings. New York: Springer-Verlag (2004): 274. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. MAPLE for n from 2 to 563 do if(ithprime(n)^2 mod 48 = 25) then print(ithprime(n)) fi od. # Gary Detlefs, Dec 29 2011 MATHEMATICA Select[Prime @ Range[2, 105], JacobiSymbol[2, # ] == -1 &] (* Robert G. Wilson v, Dec 15 2005 *) Select[Union[8Range[100] - 5, 8Range[100] - 3], PrimeQ[#] &] (* Alonso del Arte, May 22 2016 *) Select[Prime[Range[150]], MemberQ[{3, 5}, Mod[#, 8]]&] (* Harvey P. Dale, Mar 02 2022 *) PROG (PARI) is(n)=isprime(n) && (n%8==3 || n%8==5) \\ Charles R Greathouse IV, Mar 21 2016 (Magma) [3] cat [p: p in PrimesUpTo (600) | p^2 mod 48 eq 25]; // Vincenzo Librandi, May 23 2016 CROSSREFS Cf. A001132 (complement from the odd primes), A007521 (subsequence), A038873, A226523. Sequence in context: A059644 A059646 A319041 * A175865 A001122 A152871 Adjacent sequences: A003626 A003627 A003628 * A003630 A003631 A003632 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)