Integers as Sums of Four Tetrahedral Numbers and Eight Pentatope Numbers in One Identity

Abstract

In 1850, Frederick Pollock [1] conjectured that every positive integer can be written as the sum of at most five tetrahedral numbers, where the nth tetrahedral number is given by Tn = 6n(n + 1)(n + 2). Earlier this year, Vadim Ponomarenko solved the weak version of Pollock’s conjecture [2] as Tn − Tn−1 − Tn−1 + Tn−2 = n. This solution is weak in the sense that it includes subtraction as well as addition instead of only addition. The following identity is a generalization of his solution at a = 2 and k = 1.