On Universal Sums of Polygonal Numbers

For m = 3, 4, . . . , the polygonal numbers of order m are given by pm(n) = (m−2) ` n 2 ́ +n (n = 0, 1, 2, . . . ). For positive integers a, b, c and i, j, k > 3 with max{i, j, k} > 5, we call the triple (api, bpj , cpk) universal if for any n = 0, 1, 2, . . . there are nonnegative integers x, y, z such that n = api(x)+bpj(y)+cpk(z). We show that there are only 95 candidates for universal triples (two of which are (p4, p5, p6) and (p3, p4, p27)), and conjecture that they are indeed universal triples. For many triples (api, bpj , cpk) (including (p3, 4p4, p5), (p4, p5, p6) and (p4, p4, p5)), we prove that any nonnegative integer can be written in the form api(x)+bpj(y)+cpk(z) with x, y, z ∈ Z. We also show some related new results on ternary quadratic forms, one of which states that any nonnegative integer n ≡ 1 (mod 6) can be written in the form x + 3y + 24z with x, y, z ∈ Z. In addition, we pose several related conjectures on sums of primes and polygonal numbers.