by the same factor. All solutions arise in one of these two ways. For instance, the congruum 96 can be constructed by these formulas with and , while the congruum 216 is obtained by multiplying the smaller congruum 24 by the square number 9. An equivalent formulation of this solution, given by Bernard Frénicle de Bessy, is that for the three squares in arithmetic progression , , and , the middle number is the hypotenuse of a Pythagorean triangle and the other two numbers and are the difference and sum respectively of the triangle's two legs. The congruum itself is four times the area of the same Pythagorean triangle. The example of an arithmetic progression with the congruum 96 can be obtained in this way from a right triangle with side and hypotenuse lengths 6, 8, and 10. Relation to congruent numbers A congruent number is defined as the area of a right triangle with rational sides. Because every congruum can be obtained (using the parameterized solution) as the area of a Pythagorean triangle, it follows that every congruum is congruent. Conversely, every congruent number is a congruum multiplied by the square of a rational number. However, testing whether a number is a congruum is much easier than testing whether a number is congruent. For the congruum problem, the parameterized solution reduces this testing problem to checking a finite set of parameter values. In contrast, for the congruent number problem, a finite testing procedure is known only conjecturally, via Tunnell's theorem, under the assumption that the Birch and Swinnerton-Dyer conjecture is true. See also Automedian triangle, a triangle for which the squares on the three sides form an arithmetic progression Spiral of Theodorus, formed by right triangles whose (non-integer) sides, when squared, form an infinite arithmetic progression References External links Diophantine equations Integer sequences Squares in number theory Theorems in number theory