A059771 Second solution of x^2 = 2 mod p for primes p such that a solution exists.

0, 4, 11, 18, 23, 24, 40, 59, 41, 70, 64, 83, 65, 62, 111, 106, 105, 154, 134, 141, 179, 208, 148, 140, 219, 197, 153, 175, 149, 245, 193, 311, 186, 340, 288, 246, 348, 312, 243, 227, 418, 419, 377, 260, 292, 396, 346, 272, 368, 543, 451, 433, 379, 413, 321

Offset: 1

Author

Klaus Brockhaus, Feb 21 2001

Keywords

• nonn
• look

Comments

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 A059770 of the first solutions (Cf. A059772).

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A038873, A059770, A059772.

Programs

Maple

R:= 0: p:= 2: count:= 1:
while count < 100 do
  p:= nextprime(p);
  if NumberTheory:-QuadraticResidue(2,p)=1 then
    v:= NumberTheory:-ModularSquareRoot(2,p);
    R:= R, max(v,p-v);
    count:= count+1
  fi
od:
R; # Robert Israel, Sep 07 2023

Formula

a(n) = second (larger) 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. a(n) = 0 if x^2 = 2 mod p has one solution (only for p = 2).