be a square number itself, but did not present a satisfactory proof of this fact. If three squares , , and could form an arithmetic progression whose congruum was also a square , then these numbers would satisfy the Diophantine equations That is, by the Pythagorean theorem, they would form two integer-sided right triangles in which the pair gives one leg and the hypotenuse of the smaller triangle and the same pair also forms the two legs of the larger triangle. But if (as Fibonacci asserted) no square congruum can exist, then there can be no two integer right triangles that share two sides in this way. Areas of right triangles Because the congrua are exactly the numbers that are four times the area of a Pythagorean triangle, and multiplication by four does not change whether a number is square, the existence of a square congruum is equivalent to the existence of a Pythagorean triangle with a square area. It is this variant of the problem that Fermat's proof concerns: he shows that there is no such triangle. In considering this problem, Fermat was inspired not by Fibonacci but by an edition of Arithmetica by Diophantus, published in a translation into French in 1621 by Claude Gaspar Bachet de Méziriac. This book described various special right triangles whose areas had forms related to squares, but did not consider the case of areas that were themselves square. By rearranging the equations for the two Pythagorean triangles above, and then multiplying them together, one obtains the single Diophantine equation which can be simplified by introducing a new variable to Conversely, any three positive integers obeying the equation lead to a square congruum: for these numbers, the squares , , and form an arithmetic progression with congruum , which is a square itself. Thus, the solvability of is equivalent to the existence of a square congruum. But, if Fermat's Last Theorem had a counterexample for the exponent , an integer solution to the equation , then squaring one of the three numbers in the counterexample would give three numbers that solve the equation . Therefore, Fermat's proof that no Pythagorean triangle has a square area implies the truth of the exponent- case of Fermat's Last Theorem. Another equivalent formulation of the same problem involves congruent numbers, the numbers that are areas of right triangles whose three sides are all rational numbers. By multiplying