5 M ay 2 00 9 Preprint , arXiv : 0905 . 0635 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 (one of which is (p4, p5, p6)), and conjecture that they are indeed universal triples. By using the theory of ternary quadratic forms, we prove that for many triples (api, bpj , cpk) (including the triples (p3, p4, p5), (p4, p5, p6) and (p4, p4, p5)) any nonnegative integer can be written in the form api(x)+ bpj(y)+ cpk(z) with x, y, z ∈ Z. We also pose several related conjectures on sums of primes and polygonal numbers, one of which states that for any m = 5, 6, 7, . . . with m 6≡ 2 (mod 8) all sufficiently large odd integers can be written in the form p + 2pm(x) with p a prime and x an integer.