×

On the property \(P_{-1}\). (English) Zbl 1137.11020

S. P. Mohanty and A. M. S. Ramasamy proved [J. Number Theory 18, 356–359 (1984; Zbl 0534.10012)] that no other integer can be added to the set \(\{1, 5, 10\}\) such that the product of any two numbers from the new set minus one (property \(P_{-1}\)) is a perfect square. The author gives an alternative proof considering two simultaneous quadratic Diophantine equations which are equivalent to those investigated by Mohanty and Ramasamy. By carefully transforming the variables he changes the two quadratics into quartic equations where the finitely many solutions can be found in the standard textbooks.

MSC:

11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 0534.10012
PDFBibTeX XMLCite
Full Text: EuDML