Pollock's conjectures
Pollock's conjectures are two closely related unproven[1] conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock,[2][3] 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.
- Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most five tetrahedral numbers.[4]
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.[4]
- Pollock octahedral numbers conjecture: Every positive integer is the sum of at most seven octahedral numbers.[3] This conjecture has been proven for all but finitely many positive integers.[5]
- Polyhedral numbers conjecture: Every positive integer is the sum of at most 5 tetrahedral numbers, or the sum of at most 9 cube numbers, or the sum of at most 7 octahedral numbers, or the sum of at most 22 dodecahedral numbers, or the sum of at most 15 icosahedral numbers.[6] The cube numbers case was established from 1909 to 1912 by Wieferich[7] and A. J. Kempner.[8]
References
- Deza, Elena; Deza, Michael (2012). Figurate Numbers. World Scientific. ISBN 9789814355483.
- 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.
- 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.
- 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.
- Kim, Hyun Kwang (2002-06-12). "On regular polytope numbers". Proceedings of the American Mathematical Society. American Mathematical Society (AMS). 131 (1): 65–75. doi:10.1090/s0002-9939-02-06710-2. ISSN 0002-9939.
- 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.
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