New Theorem on Primitive Pythagorean Triples

Abstract

As a result of our survey on primitive Pythagorean triples, we were able to prove the following theorem: All primitive Pythagorean triples can be generated by almost one parametera , satisfyinga > 2 +1. Furthermore, a is either an integer or of the form h a = g where g and h (> 1) are relatively prime numbers. The proof of the theorem can be briefly outlined as follows: Taking z = y + p for some p ³ 1, z 2 = y 2 + x2 can be put into the form 2 2 1 1

+ =

+ y x y p If p x a = , then the above equation can be put into the form ( )2 2 2 1+b = 1+a b ........................................................................ (1), where 2 1 = a 2 -1 b . Then the above equation can be reduced into 2 2 2 2 2 2 1 2 1 1 a a a +

- =

- + . In order to generate primitive triples, the above equation has to be multiplied by 4 if a is even and h 4 if h a = g . Now we are able to generate all the primitive Pythagorean triples if a satisfies the conditions of our theorem and 2 a 2 -1 is reduced to cancel 2 in the denominator whenever necessary. The condition a > 2 +1 and a is either integer or of the form = (h > 1) h a g with g and h are relatively prime odd be imposed after a careful study of the equation . In conclusion, an algorithm can be developed to determine p and y so that (( y + p), y, x) is a primitive Pythagorean triple in the order x < y < y + p for given x. A new theorem on primitive Pythagorean triples is found and it may be useful in understanding the Fermat’s Last Theorem.