for in Euclid's formula. There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longest leg differ by exactly two. They are all primitive, and are obtained by putting in Euclid's formula. More generally, for every integer , there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by . They are obtained by putting in Euclid's formula. There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 20 + 21 = 29; these are generated by Euclid's formula when is a convergent to . For each natural number , there exist Pythagorean triples with different hypotenuses and the same area. For each natural number , there exist at least different primitive Pythagorean triples with the same leg , where is some natural number (the length of the even leg is 2mn, and it suffices to choose with many factorizations, for example , where is a product of different odd primes; this produces at least different primitive triples). For each natural number , there exist at least different Pythagorean triples with the same hypotenuse. If is a prime power, there exists a primitive Pythagorean triple if and only if the prime has the form ; this triple is unique up to the exchange of a and b. More generally, a positive integer is the hypotenuse of a primitive Pythagorean triple if and only if each prime factor of is congruent to modulo ; that is, each prime factor has the form . In this case, the number of primitive Pythagorean triples with is , where is the number of distinct prime factors of . There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse and the sum of the legs . According to Fermat, the smallest such triple has sides ; ; and . Here and . This is generated by Euclid's formula with parameter values and . There exist non-primitive Pythagorean triangles with integer altitude from the hypotenuse. Such Pythagorean triangles are known as decomposable since they can be split along this altitude into two separate and smaller Pythagorean triangles. Geometry of Euclid's formula Rational points on a unit circle Euclid's formula for a Pythagorean triple can be understood in terms of the geometry of rational points on the unit circle . In fact, a point in the Cartesian plane