A334819 Largest quadratic nonresidue modulo n (with n >= 3).

2, 3, 3, 5, 6, 7, 8, 8, 10, 11, 11, 13, 14, 15, 14, 17, 18, 19, 20, 21, 22, 23, 23, 24, 26, 27, 27, 29, 30, 31, 32, 31, 34, 35, 35, 37, 38, 39, 38, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 51, 53, 54, 55, 56, 56, 58, 59, 59, 61, 62, 63, 63, 65, 66, 67

Offset: 3

Author

Peter Schorn, May 12 2020

Keywords

• nonn
• easy

Comments

The largest nonnegative integer less than n which is not a square modulo n.

If p is a prime congruent 3 modulo 4 then a(p) = p-1 since -1 is not a quadratic residue for such primes.

Examples

			The squares modulo 3 are 0 and 1. Therefore a(3) = 2. The nonsquares modulo 4 are 2 and 3 which makes a(4) = 3. Modulo 5 we have 0, 1 and 4 as squares giving a(5) = 3. 0, 1 and 4 are also the squares modulo 6 resulting in a(6) = 5. Since 10007 is a prime of the form 4*k + 3, a(10007) = 10006.
		

Links

• Robert Israel, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A020649, A047210 (the largest square modulo n), A192450 (a(n)=n-1).

Programs

Maple

f:= proc(n) local k;
  for k from n-1 by -1 do if numtheory:-msqrt(k,n)=FAIL then return k fi
  od
end proc:
map(f, [$3..100]); # Robert Israel, May 14 2020

Mathematica

a[n_] := Module[{r}, For[r = n-1, r >= 1, r--, If[!IntegerQ[Sqrt[Mod[r, n]] ], Return[r]]]];
a /@ Range[3, 100] (* Jean-François Alcover, Aug 15 2020 *)

PARI

a(n) = forstep(r = n - 1, 1, -1, if(!issquare(Mod(r, n)), return(r)))