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Certain Diophantine Equations Linear In One Unknown

Published online by Cambridge University Press:  20 November 2018

W. H. Mills
W. H. Mills*
Affiliation:
Yale University
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1. Introduction. A. Brauer and R. Brauer (2) and Barnes (1) (following a method of Mordell (6)) have solved the Diophantine equation x2+y2+c = xyz subject to the condition (x, y) = 1. Independently, but using the same methods, I treated (4) the equation

x2+y2+ax+ay+l = xyz,

and subsequently (5) gave a method of obtaining all integral solutions of

x2±y2+ax+by+c = xyz,

thereby generalizing (2), (1), and (4).

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 8 , 1956 , pp. 5 - 12
Copyright
Copyright © Canadian Mathematical Society 1956

References

1. Barnes, E. S., On the Diophantine equation x2+y2+c = xyz, J. London Math. Soc, 28 (1953), 242–244.Google Scholar
2. Brauer, A. and Brauer, R., Lösung der Aufgabe 46, Angelegenheiten d. Deutschen Mathem.- Vereinigung, (1927), 90–92 (Jahresbericht d. Deutschen Mathem.-Vereinigung, 86).Google Scholar
3. Goldberg, Karl, Newman, Morris, Straus, E. G., and Swift, J. D., The representation of integers by binary quadratic rational forms, Archiv der Mathematik, 5 (1954), 12–18.Google Scholar
4. Mills, W. H., A system of quadratic Diophantine equations, Pacific J. Math., 3 (1953), 209–220.Google Scholar
5. Mills, W. H., A method for solving certain Diophantine equations, Proc. Amer. Math. Soc, 5 (1954), 473–475.Google Scholar
6. Mordell, L. J., The congruence ax3 + by3 + c ≡ 0 (mod xy), and integer solutions of cubic equations in three variables, Acta Math., 88 (1952), 77–83.Google Scholar