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%I M2472 #99 Sep 08 2022 08:44:32
%S 3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163,
%T 173,179,181,197,211,227,229,251,269,277,283,293,307,317,331,347,349,
%U 373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557,563
%N Primes p == +- 3 (mod 8), or, primes p such that 2 is not a square mod p.
%C Complement of A038873 relative to A000040.
%C Also primes p such that p divides 2^((p-1)/2) + 1. - _Cino Hilliard_, Sep 04 2004
%C Primes p such that p^2 == 25 (mod 48), n > 1. - _Gary Detlefs_, Dec 29 2011
%C 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
%C Except for the initial 3, these are the primes p such that Fibonacci(p) mod 6 = 5. - _Gary Detlefs_, May 26 2014
%C Inert rational primes in the field Q(sqrt(2)). - _N. J. A. Sloane_, Dec 26 2017
%C 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
%C For the proofs or the comments by Cino Hilliard and Marina Ibrishimova, see link below. - _Robert Israel_, Apr 24 2019
%D 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.
%D Ronald S. Irving, Integers, Polynomials, and Rings. New York: Springer-Verlag (2004): 274.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Ray Chandler, <a href="/A003629/b003629.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Robert Israel, <a href="/A003629/a003629.pdf">Proof of comments by Hilliard (also conjectured by Detlefs) and Ibrishimova</a>
%H <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>
%p 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
%t Select[Prime @ Range[2, 105], JacobiSymbol[2, # ] == -1 &] (* _Robert G. Wilson v_, Dec 15 2005 *)
%t Select[Union[8Range[100] - 5, 8Range[100] - 3], PrimeQ[#] &] (* _Alonso del Arte_, May 22 2016 *)
%t Select[Prime[Range[150]],MemberQ[{3,5},Mod[#,8]]&] (* _Harvey P. Dale_, Mar 02 2022 *)
%o (PARI) is(n)=isprime(n) && (n%8==3 || n%8==5) \\ _Charles R Greathouse IV_, Mar 21 2016
%o (Magma) [3] cat [p: p in PrimesUpTo (600) | p^2 mod 48 eq 25]; // _Vincenzo Librandi_, May 23 2016
%Y Cf. A001132 (complement from the odd primes), A007521 (subsequence), A038873, A226523.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, _Mira Bernstein_
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