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A333669 The smallest nontrivial quadratic residue modulo n.

%I A333669 #27 Jun 14 2022 15:39:08
%S A333669 4,3,2,4,4,4,3,4,3,2,4,4,2,4,4,4,4,3,2,4,4,3,4,4,4,4,2,4,3,2,4,4,3,4,
%T A333669 3,4,2,4,4,4,4,2,2,4,2,4,4,4,4,4,4,4,4,4,3,4,3,2,4,4,4,3,4,4,3,4,2,4,
%U A333669 2,3,4,4,4,3,2,4,4,2,3,4,4,4,4,4
%N A333669 The smallest nontrivial quadratic residue modulo n.
%C A333669 The values are 2, 3 and 4. If 2 is a square modulo n (see A057126) the value is 2. Otherwise, if 3 is a square modulo n (see A057125) the value is 3. If neither 2 or 3 are a square modulo n the value is 4.
%C A333669 Dedicated to Urs Meyer at the occasion of his 60th birthday.
%H A333669 Robert Israel, <a href="/A333669/b333669.txt">Table of n, a(n) for n = 5..10000</a>
%e A333669 The squares modulo 5 are 1 and 4, therefore a(5) = 4.
%e A333669 Modulo 6 the squares are 1, 3 and 4 which makes a(6) = 3.
%e A333669 a(7) = 2 since 2 == 3^2 (mod 7).
%p A333669 f:= proc(n) uses numtheory; if quadres(2,n)=1 then 2 elif quadres(3,n)=1 then 3 else 4 fi end proc:
%p A333669 map(f, [$5..100]); # _Robert Israel_, Sep 15 2020
%t A333669 qrQ[m_, n_] := Module[{k}, Reduce[Mod[m-k^2, n]==0, k, Integers] =!= False];
%t A333669 a[n_] := If[qrQ[2, n], 2, If[qrQ[3, n], 3, 4]];
%t A333669 a /@ Range[5, 100] (* _Jean-François Alcover_, Oct 25 2020 *)
%o A333669 (PARI) a(n) = if(issquare(Mod(2,n)),2,issquare(Mod(3,n)),3,4)
%Y A333669 Cf. A057126 for the n where the value is 2 and A057125 for the n where the value is 3 if n was not in A057126.
%K A333669 nonn,easy
%O A333669 5,1
%A A333669 _Peter Schorn_, May 07 2020