Pollock's conjectures explained

Pollock's conjectures are closely related conjectures in additive number theory.^[1] They were first stated in 1850 by Sir Frederick Pollock,^[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, ..., of 241 terms, with 343,867 conjectured to be the last such number.

• 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.^[3]

• 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^[4] and A. J. Kempner.^[5]

• 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.^[6]

Notes and References

1. Book: Dickson, L. E. . History of the Theory of Numbers, Vol. II: Diophantine Analysis . June 7, 2005 . Dover . 0-486-44233-0 . 22–23 . Leonard Eugene Dickson.
2. Frederick Pollock . Sir Frederick Pollock, 1st Baronet . On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders . Abstracts of the Papers Communicated to the Royal Society of London . 5 . 1850 . 922–924 . 111069 .
3. Elessar Brady. Zarathustra. 1509.04316. 10.1112/jlms/jdv061. 1. Journal of the London Mathematical Society. 3455791. 244–272. Second Series. Sums of seven octahedral numbers. 93. 2016. 206364502 .
4. Wieferich . Arthur . Arthur Wieferich . 1909 . Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt . Mathematische Annalen . de . 66 . 1 . 95–101 . 10.1007/BF01450913 . 121386035.
5. Kempner . Aubrey . 1912 . Bemerkungen zum Waringschen Problem . Mathematische Annalen . de . 72 . 3 . 387–399 . 10.1007/BF01456723 . 120101223.
6. Kureš . Miroslav . 2023-10-27 . A Proof of Pollock’s Conjecture on Centered Nonagonal Numbers . The Mathematical Intelligencer . en . 10.1007/s00283-023-10307-0 . 0343-6993.