On the number of primitive representations of integers as sums of squares

Abstract

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity.

The linking number in systems with Periodic Boundary Conditions

“…Recall that r 2 (n) and r 3 (n) were used in Section 5.1 and will also be used in Section 6 in the proof of convergence. Analytic expressions for these functions can be found in [63,64,65,66,67]. More precisely, Jacobi [66] proved that…”

“…where c 3 (n) = 1 if n = 0 mod 4 or c 3 (n) = 0 if n = 0 mod 4, [67]. Moreover, r 3 (n) and R 3 (n) are related by the following [63]:…”

Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

“…A seminal breakthrough in the theory of numbers is the determination by Gauss [1] of the number of representations ( ) 3 r n of an integer n as a sum of three squares 2 2 2 x y z n + + = counting zeros, permutations and sign changes (e.g. Dickson [2], Preface, pp.…”

“…ix, x). A very explicit modern expression for this counting function is given in Cooper and Hirschhorn [3], Lemma 4, Equation (3.1), and Theorem 3, Equation (1.27), (1.28). Note that the latter result has only been obtained quite recently by Hirschhorn and Sellers [4].…”

Cubes in {0,1,...,n}3

“…> ExtendList:=proc(n,N,L::list,mm,nn,Orb::array) > local i,sol,nsol,nel,ttpcube,orb::array, > LL::list,tnel,NL::list,cio,C,pC,k,kvalues,m,exception,abg,NN; > NN:=floor((N-1)/2); > orb:=array(1..2*NN+1); > nel:=nops(L); > LL:=L; > k:=sqrt(mm^2-mm*nn+nn^2); > m:=n*k; > if m<=N then > sol:=abcsol(n);nsol:=nops(sol); > tnel:=nel;orb:=Orb; > for i from 1 to nsol do > C:=findpar(sol[i] [1],sol[i] [2],sol[i] [3],mm,nn);pC:=tmttopqcube(C); > kvalues:=fourkvalues(pC);exception:=evalb (…”