Waring's problem for pyramidal numbers

Yue Fan Deng; Chen Ning Yang

Waring's problem for pyramidal numbers

Yue Fan Deng, Chen Ning Yang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

It has been proved that every positive integer is expressible as a sum of no more than eight pyramidal numbers P(m) = (m - 1)m(m + 1) 6. This paper reports on a computer calculation of the partition of integers from n = 1 to 10⁹ into pyramidal numbers. We find that no integer ≤10⁹ needs more than five pyramidal numbers for its partition, and only 241 of them do need five. We define J(n) as the least number of pyramidal numbers to partition n, and N_k(n) as the number of positive integers l less than or equal to n for which J(l) = k. Based on our numerical results we make conjectures about the asymptotic form of N_k(n) for n → ∞.

Original language	English (US)
Pages (from-to)	277-283
Number of pages	7
Journal	Science in China (Scientia Sinica) Series A
Volume	37
Issue number	3
State	Published - Mar 1994

Keywords

asymptotic form
parallel computing
Waring's problem

ASJC Scopus subject areas

Mathematics(all)
Physics and Astronomy(all)

Cite this

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AB - It has been proved that every positive integer is expressible as a sum of no more than eight pyramidal numbers P(m) = (m - 1)m(m + 1) 6. This paper reports on a computer calculation of the partition of integers from n = 1 to 109 into pyramidal numbers. We find that no integer ≤109 needs more than five pyramidal numbers for its partition, and only 241 of them do need five. We define J(n) as the least number of pyramidal numbers to partition n, and Nk(n) as the number of positive integers l less than or equal to n for which J(l) = k. Based on our numerical results we make conjectures about the asymptotic form of Nk(n) for n → ∞.

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Waring's problem for pyramidal numbers

Abstract

Keywords

ASJC Scopus subject areas

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