2009/11/06 by Oh, Byeong-Kweon
#11H12 #11H20 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0911.1181
For any m≥3, every integer of the form pm(x)=\frac(m-2)x2-(m-4)x2 with x ∈ \z is said to be a generalized m-gonal number. Let a≤ b≤ c be positive integers. For every non negative integer n, if there are integers x,y,z such that n=apk(x)+bpk(y)+cpk(z), then the quadruple (k;a,b,c) is said to be \it universal. Sun gave in \cites1 all possible quadruple candidates that are universal and proved some quadruples to be universal (see also \citegs). He remains the following quadruples (5,1,1,k) for k=6,8,9,10, (5,1,2,8), and (5,1,3,s) for 7 ≤ s ≤ 8 as candidates and conjectured the universality of them. In this article we prove that the remaining 7 quadruples given above are, in fact, universal.