A method is given for generating families of non-congruent numbers with arbitrarily many prime factors. We then use this method to construct an infinite set of new families of these numbers with prime factors of the form 8k + 3.

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A method is given for constructing congruent numbers with three prime factors of the form 8k C 3. A family of such numbers is given for which the Mordell-Weil rank of their associated elliptic curves equals 2, the maximal rank and expected rank for a congruent number curve of this type.