{{Short description|Conjectures in additive number theory}} '''Pollock's conjectures''' are closely related conjectures in additive number theory.<ref name="Dickson">{{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}}</ref> They were first stated in 1850 by Sir Frederick Pollock,<ref name="Dickson" /><ref name=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 }}</ref> 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.<ref name="mathworld">{{Mathworld|title=Pollock's Conjecture|id=PollocksConjecture}}</ref>
*'''Pollock octahedral numbers conjecture''': Every positive integer is the sum of at most 7 octahedral numbers. This conjecture has been proven for all sufficiently large numbers. Namely, every number greater than <math>e^{10^7}\approx6.59\cdot10^{4342944}</math> is sufficiently large.<ref>{{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 }}</ref> *'''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<ref>{{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}}</ref> and A. J. Kempner.<ref>{{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}}</ref> *'''Pollock icosahedral and dodecahedral numbers conjectures''': Every positive integer is the sum of at most 13 icosahedral numbers. Every positive integer is the sum of at most 21 dodecahedral numbers. These two conjectures are corrected and confirmed true in 2025. <ref>{{cite journal |last1=Basak |first1=Debmalya |last2=Dong|first2=Anji|last3=Saettone|first3=Katerina|last4=Zaharescu|first4=Alexandru|year=2025 |title=Representations as Sums of Icosahedral and Dodecahedral Numbers: Proof of Pollock’s Conjectures|journal=International Mathematics Research Notices |language=en |volume=2025 |issue=13 |doi=10.1093/imrn/rnaf180}}</ref> *'''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.<ref>{{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|url-access=subscription }}</ref>
==See also== *Centered polygonal number theorem
==References== {{reflist}}
Category:Conjectures Category:Unsolved problems in number theory Category:Figurate numbers Category:Additive number theory
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