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
[edit]- 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 sufficiently large numbers. Namely, every number greater than is sufficiently large.[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 icosahedral and dodecahedral numbers conjectures: Every positive integer is the sum of at most 13 icosahedral numbers. Every positive integer is the sum of at most 21 dodecahedral numbers.
These two conjectures are corrected and confirmed true in 2025. [7] The authors proved that every positive integer is the sum of at most 15 icosahedral numbers, and 15 is the least integer with this property. Moreover, every positive integer is the sum of at most 22 dodecahedral numbers, and 22 is the least integer with this property. Notice that the results are slightly different from Pollock's original conjectures. One can check that the number 95 can be written as a sum of 15 icosahedral numbers, but cannot be written as a sum of at most 14 icosahedral numbers. Similarly, 79 is the counterexample to Pollock's conjecture on dodecahedral numbers.
- 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.[8]
See also
[edit]References
[edit]- ^ a b Dickson, L. E. (June 7, 2005). History of the Theory of Numbers, Vol. II: Diophantine Analysis. Dover. pp. 22–23. ISBN 0-486-44233-0.
- ^ Frederick Pollock (1850). "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: 922–924. JSTOR 111069.
- ^ Weisstein, Eric W. "Pollock's Conjecture". MathWorld.
- ^ Elessar Brady, Zarathustra (2016). "Sums of seven octahedral numbers". Journal of the London Mathematical Society. Second Series. 93 (1): 244–272. arXiv:1509.04316. doi:10.1112/jlms/jdv061. MR 3455791. S2CID 206364502.
- ^ Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen (in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID 121386035.
- ^ Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID 120101223.
- ^ Basak, Debmalya; Dong, Anji; Saettone, Katerina; Zaharescu, Alexandru (2025). "Representations as Sums of Icosahedral and Dodecahedral Numbers: Proof of Pollock's Conjectures". International Mathematics Research Notices. 2025 (13). doi:10.1093/imrn/rnaf180.
- ^ Kureš, Miroslav (2023-10-27). "A Proof of Pollock's Conjecture on Centered Nonagonal Numbers". The Mathematical Intelligencer. 46 (3): 234–235. doi:10.1007/s00283-023-10307-0. ISSN 0343-6993.
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