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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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 $n$ less than $40$ billion ($40$B) but greater than $343,867$ 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, $1$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 $O(n)$ compared with that of the old, $O(n^{5/3}\log n)$.


References [Enhancements On Off] (What's this?)

  • 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 $1000000$ 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

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