A059771 -id:A059771 - cp's OEIS frontend

A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

0, 7, 7, 23, 17, 47, 31, 79, 0, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 0, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 0, 47, 73, 881, 1847, 967, 0, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 0, 3023, 1567, 191, 0, 71

Offset: 2

Klaus Brockhaus, Feb 21 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^2 = 2 iff n^2-2 has a prime factor > n; n is a solution mod p of x^2 = 2 iff p is a prime factor of n^2-2 and p > n. n^2-2 has at most one prime factor > n, consequently such a factor is the only prime p such that n is a solution mod p of x^2 = 2. For n such that n^2-2 has no prime factor > n (the zeros in the sequence), cf. A060515.

			a(11) = 17, since 11 is a solution mod 17 of x^2 = 2 and 11 is not a solution mod p of x^2 = 2 for primes p < 17. Although 11^2 = 2 mod 7, prime 7 is excluded because 7 < 11 and 11 = 4 mod 7.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A038873, A059770, A059771, A060515.

f:= proc(n) local P;
  P:= select(`>`,numtheory:-factorset(n^2-2),n);
  if P = {} then 0 else min(P) fi
end proc:
map(f, [$2..100]); # Robert Israel, Feb 23 2016

a[n_] := Module[{P}, P = Select[FactorInteger[n^2 - 2][[All, 1]], # > n&]; If[P == {}, 0, Min[P]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 10 2019, from Maple *)

If n^2-2 has a (unique) prime factor p > n, then a(n) = p, else a(n) = 0.

Offset corrected by R. J. Mathar, Aug 21 2009

A059770 First solution of x^2 = 2 mod p for primes p such that a solution exists.

Original entry on oeis.org

0, 3, 6, 5, 8, 17, 7, 12, 32, 9, 25, 14, 38, 51, 16, 31, 46, 13, 57, 52, 20, 15, 85, 99, 22, 60, 110, 96, 132, 66, 120, 26, 167, 19, 79, 137, 53, 97, 188, 206, 21, 30, 80, 203, 187, 91, 157, 249, 201, 34, 142, 166, 222, 194, 296, 94, 67, 36, 283, 324, 27, 102, 113, 73

Offset: 1

Klaus Brockhaus, Feb 21 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. For p > 2: If x^2 = 2 has a solution mod p, then it has exactly two solutions and their sum is p; i is a solution mod p of x^2 = 2 iff p-i is a solution mod p of x^2 = 2. No integer occurs more than once in this sequence. Moreover, no integer (except 0) occurs both in this sequence and in sequence A059771 of the second solutions (Cf. A059772).

			a(6) = 17, since 41 is the sixth term of A038873, 17 and 24 are the solutions mod 41 of x^2 = 2 and 17 is the smaller one.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
K. Matthews, Finding square roots mod p by Tonelli's algorithm
R. Chapman, Square roots modulo a prime

Cf. A038873, A059771, A059772.

fQ[n_] := MemberQ[{1, 2, 7}, Mod[n, 8]]; f[n_] := PowerMod[2, 1/2, n]; f@ Select[ Prime[Range[135]], fQ] (* Robert G. Wilson v, Oct 18 2011 *)

a(n) = first (least) solution of x^2 = 2 mod p, where p is the n-th prime such that x^2 = 2 mod p has a solution, i.e. p is the n-th term of A038873.

cp's OEIS Frontend

A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

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