Almost universal ternary sums of pentagonal numbers

Abstract

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $a{P}_5(x)+2^{r}b{P}_5(y)+2^{s}c{P}_5(z)$ represents all but finitely many positive integers.

arXiv comment25 pages