arXiv : 0906 . 2450 ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS

For m = 3, 4, . . . those pm(x) = (m − 2)x(x − 1)/2 + x with x ∈ Z are called generalized m-gonal numbers. Recently the second author studied for what values of positive integers a, b, c the sum ap5 + bp5 + cp5 is universal over Z (i.e., any n ∈ N = {0, 1, 2, . . . } has the form ap5(x) + bp5(y) + cp5(z) with x, y, z ∈ Z). In this paper we proved that p5 + bp5 + 3p5 (b = 1, 2, 3, 4, 9) and p5 +2p5 +6p5 are universal over Z; this partially confirms Sun’s conjecture on ap5+bp5+cp5. Sun also conjectured that any n ∈ N can be written as p3(x) + p5(y) + p11(z) and 3p3(x)+ p5(y) + p7(z) with x, y, z ∈ N; in contrast we show that p3 + p5 + p11 and 3p3 + p5 + p7 are universal over Z.