A020649 Least quadratic nonresidue modulo n (with n >= 3).

2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 5, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 5, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 3

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is the smallest q such that the congruence x^2 == q (mod n) has no solution 0 <= x < n, for n > 2. Note that a(n) is a prime. If n is an odd prime p, then a(p) is the smallest base b such that b^((p-1)/2) == -1 (mod p), see A053760. - Thomas Ordowski, Apr 24 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 3..10000
• Eric Weisstein's World of Mathematics, Quadratic Nonresidue

Crossrefs

Cf. A053760.

Programs

Mathematica

a[n_] := Min @ Complement[Range[n - 1], Mod[Range[n/2]^2, n]]; Table[a[n], {n, 3, 110}] (* Amiram Eldar, Oct 29 2020 *)

PARI

residue(n,m)={local(r);r=0;for(i=0,floor(m/2),if(i^2%m==n,r=1));r}
A020649(n)={local(r,m);r=0;m=0;while(r==0,m=m+1;if(!residue(m,n),r=1));m} \\ Michael B. Porter, Apr 30 2010

Formula

a(prime(n)) = A053760(n) for n > 1. - Thomas Ordowski, Apr 24 2019