Universal mixed sums of generalized 4-and 8-gonal numbers

Abstract

An integer of the form P-m(x) = (m-2)x(2)-(m-4)x/2 for an integer x is called a generalized m-gonal number. For positive integers alpha(1),..., alpha(u) and beta(1),..., beta(v), a mixed sum Phi = alpha P-1(4)(x(1)) + ... + alpha P-u(4)(x(u)) + beta P-1(8)(y(1)) + ... + beta P-v(8)(y(v)) of generalized 4- and 8-gonal numbers is called universal if Phi = N has an integer solution for every nonnegative integer N. In this paper, we prove that there are exactly 1271 proper universal mixed sums of generalized 4- and 8-gonal numbers. Furthermore, the "61-theorem" is proved, which states that an arbitrary mixed sum of generalized 4- and 8-gonal numbers is universal if and only if it represents the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 30, 60, and 61.