Unsolved problems in number theory | Figurate numbers | Additive number theory | Conjectures

Pollock's conjectures

Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, 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. The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (sequence in the OEIS) of 241 terms, with 343867 being almost certainly the last such number. * Pollock octahedral numbers conjecture: Every positive integer is the sum of at most seven octahedral numbers. This conjecture has been proven for all but finitely many positive integers. * 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) (Wikipedia).

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Related pages

Additive number theory | Icosahedral number | Octahedral number | Vertex (geometry) | Dodecahedral number | Fermat polygonal number theorem | Conjecture | Platonic solid | Tetrahedral number