Pollock's conjectures

Pollock's conjectures are closely related conjectures in additive number theory.^[1] They were first stated in 1850 by Sir Frederick Pollock,^[1][2] 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

• Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most 5 tetrahedral numbers.

The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (sequence A000797 in the OEIS) of 241 terms, with 343,867 conjectured to be the last such number.^[3]

• Pollock octahedral numbers conjecture: Every positive integer is the sum of at most 7 octahedral numbers.

This conjecture has been proven for all but finitely many positive integers.^[4]

• Pollock cube numbers conjecture: Every positive integer is the sum of at most 9 cube numbers.

The cube numbers case was established from 1909 to 1912 by Wieferich^[5] and A. J. Kempner.^[6]

• Pollock centered nonagonal numbers conjecture: Every positive integer is the sum of at most 11 centered nonagonal numbers.

This conjecture was confirmed as true in 2023.^[7]