New methods for obtaining new families of congruent numbers (original) (raw)

Notes on Number Theory and Discrete Mathematics

In this note, we introduce elementary methods for obtaining new families of congruent numbers (CNs). By our methods, we can produce other CNs when one or two CNs are given. Also, we use some of the Pell equations (PEs) for getting some families of CNs. Up to now, it is not exactly determined which prime numbers of the form p = 8k + 1 are CNs. Among other things, we also introduce two simple methods to find some CNs of the forms p ≡ 1 (mod 8) and 2p where p is a prime number. By non-CNs and our methods, we also obtain some Diophantine equations (especially of degree 4), which have no positive solutions. In the end, we obtain a result on Heron triangles.

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