Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T19:02:12.098Z Has data issue: false hasContentIssue false

The Diophantine equation x2+3 = yn

Published online by Cambridge University Press:  18 May 2009

J. H. E. Cohn
Affiliation:
Department of Mathematics, RHBNC, Egham, Surrey TW20 0EX.
Rights & Permissions [Opens in a new window]

Extract

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.

Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

REFERENCES

1.Euler, L., Algebra, Volume 2.Google Scholar
2.Lebesgue, V. A., Sur l'impossibilité en nombres entiers de l'équation xm = y 2 + 1, Nouvelles Annales des Mathématiques (1) 9 (1850) 178.Google Scholar
3.Mordell, L. J., Diophantine equations (Academic Press, 1969).Google Scholar
4.Nagell, T., Verallgeminerung eines Fermatschen Satzes, Archiv der Mathematik 5 (1954), 153159.CrossRefGoogle Scholar
5.Nagell, T., Contributions to the theory of a category of Diophantine equations of the second degree with two unknowns, Nova Acta Regiae Soc. Sc. Upsaliensis (4) 16 No. 2 (1955), 138.Google Scholar
6.Nagell, T., On the Diophantine equation x 2+8D = yn, Arkiv för Matematik 3 (1955), 103112.CrossRefGoogle Scholar
7.Shorey, T. N., van der Poorten, A. J., Tijdeman, R. and Schinzel, , Applications of the Gel'fond–Baker method to diophantine equations, in Transcendence Theory: Advances and Applications, (Academic Press, 1977), 5977.Google Scholar
8.Shorey, T. N. and Tijdeman, , Exponential Diophantine equations, (Cambridge University Press, 1986).CrossRefGoogle Scholar