A General Relation Between Sums of Squares and Sums of Triangular Numbers

Abstract

Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.

Analogues of Jacobi's Two-Square Theorem: An Informal Account

“…Then Theorem 1 of Adiga, Cooper and Han [1] produces (among many such relations) the following: r (3,2) (8n + 5) = 4t (3,2) (n), r (5,1) (8n + 6) = 4t (5,1) (n), r (6,1) (8n + 7) = 4t (6,1) (n), r (5,2) (8n + 7) = 4t (5,2) (n).…”

“…Interestingly, Dickson [6] gives r (k,m) (n) for several other instances of (k, m). For each of these instances, however, Theorem 1 of Adiga, Cooper, and Han [1] is not broad enough in scope to enable any of our remaining conjectures to be proved.…”

“…where y 1 , y 2 , y 3 , and y 4 are odd, positive integers, for the cases (λ 1 , λ 2 , λ 3 , λ 4 ) = (1, 1, 1, 3), (1,3,3,3), (1, 2, 2, 3), (1,3,6,6), (1,3,4,4), (1,1,2,6), and (1,3,12,12). This paper also contains a comprehensive history of representations by similar forms that dates back to Eisenstein and Liouville.…”

Transformation Formulas for the Number of Representations of by Linear Combinations of Four Triangular Numbers

“…. }, let t(a, b, c, d; n) be the number of representations of n by 1 2 ax(x + 1) + 1 2 by(y + 1) + 1 2 cz(z + 1) + 1 2 dw(w + 1) with x, y, z, w ∈ Z. In 1832, Legendre stated that t(1, 1, 1, 1; n) = 16σ(2n + 1),…”

The relations between N(a,b,c,d;n) and t(a,b,c,d;n) and (p,k)-parametrization of theta functions

“…(6.45) Substituting (2.22)-(2.27) into (6.44) and (6.45), we find N 0 (q) = N 1 (q) = 0 and thence,L 1 (q) = 0. (6.46)Based on (6.39), (6.43) and (6.46), we get F 6 (q) = 0 and for n ∈ N, 16) into (7.1), picking out the terms involving q 3n , then replacing q 3 by q, we obtain(1,3,12, 36; 6n + 3)q n = 2If we substitute (2.10) and (2.11) into (7.3), extract the terms involving q 2n , and replace q 2 by q,…”