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 343867 being almost certainly 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]
- Pollock icosahedral numbers conjecture: Every positive integer is the sum of at most 13 icosahedral numbers.
- Pollock dodecahedral numbers conjecture: Every positive integer is the sum of at most 21 dodecahedral numbers.
References
- ↑ 1.0 1.1 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.
- ↑ Weisstein, Eric W.. "Pollock's Conjecture". http://mathworld.wolfram.com/PollocksConjecture.html.
- ↑ Elessar Brady, Zarathustra (2016). "Sums of seven octahedral numbers". Journal of the London Mathematical Society. Second Series 93 (1): 244–272. doi:10.1112/jlms/jdv061.
- ↑ Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt" (in de). Mathematische Annalen 66 (1): 95–101. doi:10.1007/BF01450913. http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D38240.
- ↑ Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem" (in de). Mathematische Annalen 72 (3): 387–399. doi:10.1007/BF01456723. http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28751.
- ↑ Kureš, Miroslav (2023-10-27). "A Proof of Pollock’s Conjecture on Centered Nonagonal Numbers" (in en). The Mathematical Intelligencer. doi:10.1007/s00283-023-10307-0. ISSN 0343-6993. https://link.springer.com/10.1007/s00283-023-10307-0.
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