Pythagorean Triples and Cryptographic Coding

This paper summarizes basic properties of PPTs and shows that each PPT belongs to one of six different classes. Mapping an ordered sequence of PPTs into a corresponding sequence of these six classes makes it possible to use them in cryptography. We pose problems whose solution would facilitate such cryptographic application. Introduction A Pythagorean triple (a, b, c) consists of positive integers that are the sides of a right triangle and thus a + b = c. Given a Pythagorean triple (a, b, c), we have other similar triples that are d(a, b, c), where d > 1. A primitive Pythagorean triple (PPT) consists of numbers that are relatively prime. Pythagorean triples have been found on cuneiform tablets of Babylon [1] and they are important in Vedic ritual and described in early geometry books of India [2]-[5] and in the works of Euclid and Diophantus. For a PPT, a,b,c cannot all be even. Also, a, b cannot both be odd and c even, because then a + b is divisible by 2, whereas c is divisible by 4. One of a and b must, therefore, be odd, and we will use the convention that b is even. Also note that the factors (c b) and (c+b) of (cb) must both be squares because they cannot have common factors other than 1 for otherwise they would not be primitive. If c+b = s and c-b=t, where s and t are different odd integers with no common factors, solving them yields: