Congruent Numbers Via the Pell Equation and its Analogous Counterpart

Abstract: The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections between Pythagorean triples and the Pell equation $x<sup>2-dy<sup>2=1$ plus its analogous counterpart $x<sup>2-dy<sup>2=-1$ which give rise to congruent numbers n with arbitrarily many prime factors.