Pollock's conjectures
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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: Let m be the number of vertices of a platonic solid “regular n-hedron” (n is 4, 6, 8, 12, or 20), then every positive integer is the sum of at most m+1 n-hedral numbers. (i.e. 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 21 dodecahedral numbers, or the sum of at most 13 icosahedral numbers)
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.
- ^ 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.
- ^ a b 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.
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