Pollock's conjectures
{{Short description|Conjectures in additive number theory}}
Pollock's conjectures are closely related conjectures in additive number theory.{{cite book |author=Dickson, L. E. |title=History of the Theory of Numbers, Vol. II: Diophantine Analysis |date=June 7, 2005 |publisher=Dover |isbn=0-486-44233-0 |pages=22–23 |author-link=Leonard Eugene Dickson}} They were first stated in 1850 by Sir Frederick Pollock,{{cite journal |author = Frederick Pollock |author-link = Sir Frederick Pollock, 1st Baronet |title = 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 |journal = Abstracts of the Papers Communicated to the Royal Society of London |volume = 5 |year = 1850 |pages = 922–924 |jstor = 111069 }} 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, ..., {{OEIS|id=A000797}} of 241 terms, with 343,867 conjectured to be the last such number.{{Mathworld|title=Pollock's Conjecture|id=PollocksConjecture}}
- 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.{{cite journal|last=Elessar Brady|first=Zarathustra|arxiv=1509.04316|doi=10.1112/jlms/jdv061|issue=1|journal=Journal of the London Mathematical Society|mr=3455791|pages=244–272|series=Second Series|title=Sums of seven octahedral numbers|volume=93|year=2016|s2cid=206364502 }}
- 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{{cite journal |last=Wieferich |first=Arthur |author-link=Arthur Wieferich |year=1909 |title=Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt |url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D38240 |journal=Mathematische Annalen |language=de |volume=66 |issue=1 |pages=95–101 |doi=10.1007/BF01450913 |s2cid=121386035}} and A. J. Kempner.{{cite journal |last=Kempner |first=Aubrey |year=1912 |title=Bemerkungen zum Waringschen Problem |url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28751 |journal=Mathematische Annalen |language=de |volume=72 |issue=3 |pages=387–399 |doi=10.1007/BF01456723 |s2cid=120101223}}
- 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.{{Cite journal |last=Kureš |first=Miroslav |date=2023-10-27 |title=A Proof of Pollock's Conjecture on Centered Nonagonal Numbers |url=https://link.springer.com/10.1007/s00283-023-10307-0 |journal=The Mathematical Intelligencer |volume=46 |issue=3 |pages=234–235 |language=en |doi=10.1007/s00283-023-10307-0 |issn=0343-6993}}
References
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Category:Unsolved problems in number theory
Category:Additive number theory
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