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 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]
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 0-486-44233-0.22-23&rft.pub=Dover&rft.date=2005-06-07&rft.isbn=0-486-44233-0&rft.au=Dickson, L. E.&rfr_id=info:sid/en.wikipedia.org:Pollock's conjectures" class="Z3988">
- ^ 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 111069.922-924&rft.date=1850&rft_id=https://www.jstor.org/stable/111069#id-name=JSTOR&rft.au=Frederick Pollock&rfr_id=info:sid/en.wikipedia.org:Pollock's conjectures" class="Z3988">
- ^ 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 206364502.244-272&rft.date=2016&rft_id=info:arxiv/1509.04316&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=3455791#id-name=MR&rft_id=https://api.semanticscholar.org/CorpusID:206364502#id-name=S2CID&rft_id=info:doi/10.1112/jlms/jdv061&rft.aulast=Elessar Brady&rft.aufirst=Zarathustra&rfr_id=info:sid/en.wikipedia.org:Pollock's conjectures" class="Z3988">
- ^ 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 121386035.95-101&rft.date=1909&rft_id=info:doi/10.1007/BF01450913&rft_id=https://api.semanticscholar.org/CorpusID:121386035#id-name=S2CID&rft.aulast=Wieferich&rft.aufirst=Arthur&rft_id=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D38240&rfr_id=info:sid/en.wikipedia.org:Pollock's conjectures" class="Z3988">
- ^ Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID 120101223.387-399&rft.date=1912&rft_id=info:doi/10.1007/BF01456723&rft_id=https://api.semanticscholar.org/CorpusID:120101223#id-name=S2CID&rft.aulast=Kempner&rft.aufirst=Aubrey&rft_id=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28751&rfr_id=info:sid/en.wikipedia.org:Pollock's conjectures" class="Z3988">
- ^ Kureš, Miroslav (2023-10-27). "A Proof of Pollock's Conjecture on Centered Nonagonal Numbers". The Mathematical Intelligencer. doi:10.1007/s00283-023-10307-0. ISSN 0343-6993.
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