A059771 - OEIS

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

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OFFSET

1,2

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).

LINKS

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

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).

EXAMPLE

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.

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

od:

R; # Robert Israel, Sep 07 2023