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The smallest nontrivial quadratic residue modulo n.
1

%I #27 Jun 14 2022 15:39:08

%S 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 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 2,3,4,4,4,3,2,4,4,2,3,4,4,4,4,4

%N The smallest nontrivial quadratic residue modulo n.

%C 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 Dedicated to Urs Meyer at the occasion of his 60th birthday.

%H Robert Israel, <a href="/A333669/b333669.txt">Table of n, a(n) for n = 5..10000</a>

%e The squares modulo 5 are 1 and 4, therefore a(5) = 4.

%e Modulo 6 the squares are 1, 3 and 4 which makes a(6) = 3.

%e a(7) = 2 since 2 == 3^2 (mod 7).

%p 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 map(f, [$5..100]); # _Robert Israel_, Sep 15 2020

%t qrQ[m_, n_] := Module[{k}, Reduce[Mod[m-k^2, n]==0, k, Integers] =!= False];

%t a[n_] := If[qrQ[2, n], 2, If[qrQ[3, n], 3, 4]];

%t a /@ Range[5, 100] (* _Jean-François Alcover_, Oct 25 2020 *)

%o (PARI) a(n) = if(issquare(Mod(2,n)),2,issquare(Mod(3,n)),3,4)

%Y 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 nonn,easy

%O 5,1

%A _Peter Schorn_, May 07 2020