Ternary Universal Sums of Generalized Pentagonal Numbers

Abstract

with x ∈ Z is said to be a generalized m-gonal number. Let a ≤ b ≤ c and k be positive integers. The quadruple (k, a, b, c) is said to be universal if for every nonnegative integer n there exist integers x, y, z such that n = ap k (x)+bp k (y)+cp k (z). Sun proved in [16] that, when k = 5 or k ≥ 7, there are only 20 candidates for universal quadruples, which he listed explicitly and which all involve only the case of pentagonal numbers (k = 5). He verified that six of the candidates are in fact universal and conjectured that the remaining ones are as well. In a subsequent paper [3], Ge and Sun established universality for all but seven of the remaining candidates, leaving only (5, 1, 1, t) for t = 6, 8, 9, 10, (5, 1, 2, 8) and (5, 1, 3, s) for s = 7, 8 as candidates. In this article, we prove that the remaining seven quadruples given above are, in fact, universal.

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

“…This completes the proof. The proofs of all remaining cases are quite similar to those of Cases (4-1) or (4)(5)(6)(7)(8)(9)(10)(11). This completes the proof of Case (4-18).…”

“…Then one may easily show that for a positive integer which is congruent to 2 modulo 6 and not of the form 2 2l`1 p8k`7q for any nonnegative integers l, k is represented by M f . The rest of the proof is quite similar to that of Case (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18).…”

“…Therefore, the equation 1`2 1t 2 1 " 3N`5 has an integer solution px 1 , y 1 , z 1 , t 1 q P Z 4 such that x 1 y 1 z 1 ı 0 pmod 3q for any nonnegative integer N except 14. The proof of Case (5)(6)(7)(8)(9)(10)(11)(12)(13)(14) follows immediately.…”

“…In [6], [9], and [11], we developed a method that determines whether or not integers in an arithmetic progression are represented by some particular ternary quadratic form. We briefly introduce this method for those who are unfamiliar with it.…”

“…As in Theorem 4.2, we may assume that uv ‰ 0. Since the proofs are quite similar to each other, we only provide, as representative cases, the proofs of Cases (5-2), (5)(6)(7)(8)(9)(10)(11)(12)(13)(14), and (5-16). Case (5-2) pα 1 , α 2 , α 3 , β 1 q " p1, 2, 5, 5q.…”