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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Integer squares with constant second difference
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by Duncan A. Buell PDF
Math. Comp. 49 (1987), 635-644 Request permission

Abstract:

The problem addressed is this: Do there exist nonconsecutive integers ${n_0},{n_1},{n_2}, \ldots ,$, such that the second differences of the squares of the ${n_i}$, are constant? Specifically, can that constant be equal to 2? A complete characterization of sequences of length four can be given. The question of whether or not sequences of length five exist is still open but the existence or nonexistence of such sequences can be described in a more algorithmic way than the simple statement of the problem.
References
  • D. Allison, On square values of quadratics, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 381–383. MR 830351, DOI 10.1017/S030500410006432X
  • E. J. Barbeau, Numbers differing from consecutive squares by squares, Canad. Math. Bull. 28 (1985), no. 3, 337–342. MR 790955, DOI 10.4153/CMB-1985-040-9
  • Douglas C. Hensley, "Sequences of squares with second difference of 2 and a problem of logic," unpublished. Leonard Lipschitz, personal correspondence.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 49 (1987), 635-644
  • MSC: Primary 11Y55; Secondary 11B83
  • DOI: https://doi.org/10.1090/S0025-5718-1987-0906196-9
  • MathSciNet review: 906196