Pollock's conjectures

Pollock's conjectures are closely related conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the Royal Society. These conjectures are a partial extension of the Fermat polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers.

Statement of the conjectures
The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., of 241 terms, with 343,867 conjectured to be the last such number.
 * Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most 5 tetrahedral numbers.

This conjecture has been proven for all but finitely many positive integers. The cube numbers case was established from 1909 to 1912 by Wieferich and A. J. Kempner. This conjecture was confirmed as true in 2023.
 * Pollock octahedral numbers conjecture: Every positive integer is the sum of at most 7 octahedral numbers.
 * Pollock cube numbers conjecture: Every positive integer is the sum of at most 9 cube numbers.
 * Pollock centered nonagonal numbers conjecture: Every positive integer is the sum of at most 11 centered nonagonal numbers.