In Search of Perfect Triangles

Abstract

Working on the perfect–triangle problem was a real mountaintop experience in mathematics. Our problem–solving path started with basic geometry, followed by a little algebra, then more geometry and more algebra. Our tour was enhanced by using a computer not to solve the problem but to explore its solution.

89.02 Some obvious facts about the primes?

“…Bertrand's postulate states that there is a prime between n and 2« (n E N, n > 2). It was proved by P. Chebyshev, see [1].…”

“…The prime number theorem states that (TT (n) log nln) -> 1 as n -> °°, where n(n) is the number of primes that do not exceed n, see [1]. Our first proposition is stated as follows.…”

89.01 Perfect, and almost perfect, Pythagorean triangles

“…In this article we are interested in integer-sided Pythagorean triangles which have the property that, numerically, the perimeter is an exact divisor of the area. For x = 1, it is known there are just two solutions (see, for example, [1]), which are called perfect Pythagorean triangles. These are (6, 8, 10) and (5, 12, 13).…”