Pythagorean theorem three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization by Pappus of Alexandria in 4 AD The
Fermat's right triangle theorem that the numbers to be squared are the difference of legs, hypotenuse, and sum of legs of a Pythagorean triangle, and that the congruum is four times the area of the same triangle. In his later work on the congruum problem, published in "The Book of Squares", Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact. If three squares formula_4, formula_5, and formula_6 could form an arithmetic progression whose congruum was also a square formula_7, then these numbers would satisfy the Diophantine equations That