A288677 Every element of Z/nZ can be expressed as a sum of no more than a(n) squares.

1, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 4, 3, 2, 2, 3, 2, 3, 2, 4, 2, 2, 2, 3, 2, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 4, 2, 3, 3, 3

Offset

2,2

Links

Matthew Conroy, Table of n, a(n) for n = 2..10000

Charles Small, Waring's problem mod n, Amer. Math. Monthly 84 (1977), no. 1, 12--25.

Formula

From Small's paper, theorem 3.1: a(n)=1 if n=2; else a(n)=2 if n != 0 mod 4 and p^2|n implies p=1 mod 4; else a(n)=3 if n!=0 mod 8; else a(n)=4.

Example

0^2 = 0 and 1^2 = 1 mod 2, so each element of Z/2Z is a square, so a(2)=1;
0^2 = 0, 1^2 = 2^2 = 1 mod 3, so 2 = 1^2 + 1^2 requires two squares to sum to 2, so a(3)=2.

Mathematica

a[n_] := Which[n == 2, 1, Mod[n, 4] != 0 && AllTrue[Select[Divisors[n] // Sqrt, IntegerQ], Mod[#, 4] == 1&], 2, Mod[n, 8] != 0, 3, True, 4];
Table[a[n], {n, 2, 140}] (* Jean-François Alcover, Jun 13 2017 *)

Prog

(PARI) c(n) = A=factor(n); ok=1; for(i=1, matsize(A)[1], if(A[i, 1]%4==3&&A[i, 2]>1, ok=0)); return(ok); wn(n) = if(n==2, 1, if(n%4>0&&c(n)==1, 2, if(n%8>0, 3, 4))); for(ii=2, 140, print1(wn(ii), ", "))

Crossrefs

Cf. A287286.

Sequence in context: A305048 A205717 A304689 * A187757 A286529 A306225

Adjacent sequences: A288674 A288675 A288676 * A288678 A288679 A288680

Keyword

nonn

Author

Matthew Conroy, Jun 13 2017

Status

approved