Some Results concerning Pythagorean Triplets

By using a computer program devised by M. Creutz, we were able to determine all Pythagoeran triplets for which z ^ 300. At this point, a distinction must must be made between P-triplets for which x9 y9 and z have no common divisor [the so-called "primitive solutions" of (1)] and P-triplets which are related to the primitive solutions by multiplication by a common integer factor k. So, if Xi9y^9 zi are relatively prime and obey (1), it is obvious that the derived triplet (kx£S ky^, kzi) will also satisfy (1). The original computer program was therefore modified to print out only the primitive solutions, and was extended up to 2 < 3000. To anticipate one of my results, the number of primitive solutions in any interval of 100 in z is approximately constant and equal to « 16. Thus there are 80 primitive solutions (PS) between 3 = 1 and 500, and 477 PS in the entire interval 1 < z < 3000. We