Decomposing 40 billion integers

by four tetrahedral numbers

Authors:
Chung-Chiang Chou and Yuefan Deng

Journal:
Math. Comp. **66** (1997), 893-901

MSC (1991):
Primary 11P05, 65Y05, 68Q25

DOI:
https://doi.org/10.1090/S0025-5718-97-00818-1

MathSciNet review:
1397442

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Abstract | References | Similar Articles | Additional Information

Abstract: Based upon a computer search performed on a massively parallel supercomputer, we found that any integer less than billion (B) but greater than can be written as a sum of four or fewer tetrahedral numbers. This result has established a new upper bound for a conjecture compared to an older one, B, obtained a year earlier. It also gives more accurate asymptotic forms for partitioning. All this improvement is a direct result of algorithmic advances in efficient memory and cpu utilizations. The heuristic complexity of the new algorithm is compared with that of the old, .

**1.**Y. Deng and C. N. Yang,*Waring's problem for pyramidal numbers,*Science in China (Series A)**37**(1994) 277-283. MR**95m:11109****2.**H. E. Salzer and N. Levine,*Table of integers not exceeding**that are not expressible as the sum of four tetrahedral numbers,*Mathematics Tables and Other Aids to Computation,**12**(1958) 141-144. MR**20:6194****3.***C. Hooley, On the representations of a number as the sum of two cubes,*Math Z.**82**(1963) 259-266. MR**27:5742**

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Additional Information

**Chung-Chiang Chou**

Affiliation:
Department of Mathematics, National ChangHua University of Education, ChangHua 50058, Taiwan

**Yuefan Deng**

Affiliation:
Center for Scientific Computing, State University of New York at Stony Brook, Stony Brook, New York 11794

DOI:
https://doi.org/10.1090/S0025-5718-97-00818-1

Keywords:
Waring's problem,
parallel computing,
asymptotic form

Received by editor(s):
February 20, 1995

Received by editor(s) in revised form:
May 22, 1995, and March 27, 1996

Article copyright:
© Copyright 1997
American Mathematical Society