Abstract
Abstract The congruent number problem. A congruent numberis a rational number qthat is the area of a right triangle, all of whose sides have rational length. We observe that if the triangle has sides a, b, and c, and if sis a rational number, then s2qis also a congruent number whose associated triangle has sides sa, sb,and sc.So it is enough to ask which square free integers nare congruent numbers. If we take cto be the length of the hypotenuse, then we are looking for square free integers nsuch that there are rational numbers a, b, csatisfying A simple algebraic calculation shows that the positive solutions to the simultaneous equations (25.1.1) are in one-to-one correspondence with the positive solutions to the equation Thus nis a congruent number if and only if (25.1.2) has a solution in positive rational numbers xand y.
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