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Sieving with sums

Published online by Cambridge University Press:  16 November 2021

Jack M. Robertson  and
William A. Webb
Jack M. Robertson
Affiliation:
Washington State University
William A. Webb
Affiliation:
Washington State University
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Extract

A number of deep theorems and unsolved problems ask questions about the relationship of the additive and multiplicative structure of the natural numbers. The famous four squares theorem of Lagrange says that if we form all possible sums of four squares we get the entire sequence of nonnegative integers. Goldbach’s conjecture states that if all possible sums of two odd primes are formed we get every even integer greater than 4.

The depth and difficulty of such problems stand in surprising contrast to the similar question: “Which numbers are not generated when all possible sums of two or more consecutive natural numbers are formed?” The answer is all powers of 2 and the proof is accessible to anyone who can sum arithmetic sequences. This result is an excellent discovery exercise for students, particularly appropriate for teacher training courses.

Type
Research Article
Information
The Mathematical Gazette , Volume 75 , Issue 472 , June 1991 , pp. 171 - 174
Copyright
Copyright © The Mathematical Association 1991

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References

1. Guy, R., Sums of consecutive integers, The Fibonacci Quarterly 20, No. 1 (1982): 36–38.Google Scholar
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3. Leveque, W.J., On representation as a sum of consecutive integers, Canad. J. Math. 4 (1950): 399–405.10.4153/CJM-1950-036-3CrossRefGoogle Scholar