the sides by a common denominator, any congruent number may be transformed into the area of a Pythagorean triangle, from which it follows that the congruent numbers are exactly the numbers formed by multiplying a congruum by the square of a rational number. Therefore, the existence of a square congruum is equivalent to the statement that the number 1 is not a congruent number. Another more geometric way of stating this formulation is that it is impossible for a square (the geometric shape) and a right triangle to have both equal areas and all sides commensurate with each other. Elliptic curve Yet another equivalent form of Fermat's theorem involves the elliptic curve consisting of the points whose Cartesian coordinates satisfy the equation The points (−1,0), (0,0), and (1,0), provide obvious solutions to this equation. Fermat's theorem is equivalent to the statement that these are the only points on the curve for which both and are rational. More generally, the right triangles with rational sides and area correspond one-for-one with the rational points with positive -coordinate on the elliptic curve . Fermat's proof During his lifetime, Fermat challenged several other mathematicians to prove the non-existence of a Pythagorean triangle with square area, but did not publish the proof himself. However, he wrote a proof in his copy of Diophantus's Arithmetica, the same copy in which he wrote that he could prove Fermat's Last Theorem. Fermat's son Clement-Samuel published an edition of this book, including Fermat's marginal notes with the proof of the right triangle theorem, in 1670. Fermat's proof is a proof by infinite descent. It shows that, from any example of a Pythagorean triangle with square area, one can derive a smaller example. Since Pythagorean triangles have positive integer areas, and there does not exist an infinite descending sequence of positive integers, there also cannot exist a Pythagorean triangle with square area. In more detail, suppose that , , and are the integer sides of a right triangle with square area. By dividing by any common factors, one can assume that this triangle is primitive and from the known form of all primitive Pythagorean triples, one can set , , and , by which the problem is transformed into finding relatively prime integers and (one of which is even) such that the area is square. For this number to be a square, its four linear factors , , , and