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]