Remarks and results on congruent numbers. (English) Zbl 0259.10010

Alter, Ronald; Curtz, Thaddeus B.; Kubota, K. K.

Proc. 3rd Southeast. Conf. Combinat., Graph Theory, Computing, Florida Atlantic Univ., Boca Raton 1972, 27-35 (1972).

A congruent number is a positive integer \(a\) such that the equations \(x^2+ay^2=z^2\) and \(x^2-ay^2=t^2\) are simultaneously soluble in positive integers \(x,y,z,t\). The authors have investigated the 608 square-free \(a\leq 1000\) with the help of a computer, whereby 198 of these numbers have been proved congruent and 136 non-congruent, the other 274 remain undecided. The results agree with the conjecture that every \(a\equiv 5,6\) or \(7\pmod 8\) is a congruent number.
For the entire collection see Zbl 0258.00003.

Reviewer: G. L. Watson

MSC:

11D09	Quadratic and bilinear Diophantine equations
11-04	Software, source code, etc. for problems pertaining to number theory

Keywords:

congruent numbers; quadratic Diophantine equations

Citations:

Zbl 0258.00003