Fermat's right triangle theorem is, by the Pythagorean theorem, they would form two integer-sided right triangles in which the pair formula_10 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. 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
Schwarz triangle function general method which works for all right angled triangles for which the two other angles are less than or equal to /4. The previous method for constructing "P", "P", ... is modified by adding an extra triangle each time an angle 3/2 arises at a vertex. The same reasoning applies to prove there is no overlapping and that the tiling covers the hyperbolic upper half plane. On the other hand, the given configuration gives rise to an arithmetic triangle group. These were first studied in . and have given rise to an extensive literature. In 1977 Takeuchi obtained a complete