Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T14:48:32.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Numbers Differing from Consecutive Squares by Squares

Published online by Cambridge University Press:  20 November 2018

E. J. Barbeau
E. J. Barbeau*
Affiliation:
University of Toronto, Toronto, M5S 1A1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.

Keywords

10B05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 3 , 01 September 1985 , pp. 337 - 342
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Baker, A. and Davenport, H., The equations 3x2 - 2 = y2 and 8x2 - 7 = z2 , Quart. J. Math. (2) 20 (1969), pp. 129–137.Google Scholar
2. Mignotte, M., Intersection des images de certaines suites récurrentes linéaires, Theor. Comp. Sci. 7 (1978), pp. 117–122.Google Scholar
3. Mordell, L.J., Diophantine Equations, (Academic, 1969).Google Scholar
4. Niven, I. and Zuckerman, H.S., An Introduction to the Theory of Numbers, (4th ed.) Wiley, 1960, 1966, 1972, 1980.Google Scholar