“Pollock's conjectures”的意思、由来-开放百科全书

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Pollock's conjectures

释义

References

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.^[3]

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 343867 being almost certainly the last such number.^[3]

Pollock octahedral numbers conjecture: Every positive integer is the sum of at most seven octahedral numbers.^[4]

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

1. ^{{cite book|title=Figurate Numbers|url=https://books.google.com/books?id=cDxYdstLPz4C|last1=Deza|first1=Elena|last2=Deza|first2=Michael|author2link=Michel Deza||publisher=World Scientific|year=2012}}
2. ^{{cite journal |author = Frederick Pollock |authorlink = 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 }}
3. ^¹{{Mathworld|title=Pollock's Conjecture|id=PollocksConjecture}}
4. ^¹ {{cite book | author = Dickson, L. E. | authorlink = Leonard Eugene Dickson | title = History of the Theory of Numbers, Vol. II: Diophantine Analysis | publisher= Dover | date = June 7, 2005 |pages=22–23 | isbn = 0-486-44233-0}}

3 : Conjectures|Figurate numbers|Additive number theory

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