Reciprocal Properties of Pythagorean triangles

TL;DR

This paper investigates specific reciprocal properties of Pythagorean triples, characterizing which triples satisfy certain reciprocal equations, and proves the uniqueness of the (3,4,5) triple for a particular property, with several non-existence results.

Contribution

It introduces the reciprocal property R(v,k,l) for Pythagorean triples and provides new theorems characterizing the existence or non-existence of such triples under various conditions.

Findings

The only Pythagorean triple with R(1,1,1) is (3,4,5).

No triples satisfy R(2,k,1) for any k.

No triples satisfy R(v,k,1) when v-1 and v+1 are twin primes.

Abstract

Let (a,b,c)be a Pythagorean triple with c being the hypotenuse length, and h being the altitude to the hypotenuse. Also, let v,k,l be positive integers with k and l being relatively prime.We say(Definition1 in this work)that the Pythagorean triple (a,b,c) has the reciprocal property R(v,k,l)if the positive integers a,b,c,v,k,and l satisfy the condition or equation, 1/a+1/b+v/h = k/l. The motivating force behind this work, is a problem that appeared in the journal, Crux Mathematicorum with Mathematical Mayhem. The said problem is Mayhem problem M390, and it appeared in the April2009 issue of the journal(see reference {1}). A solution to the same problem was published in the February 2010 issue(see {2}).Using the above definition, the probem can be stated as follows:Find all the Pythagorean triples that have the reciprocal property R(1,1,1). It turns out that the only such triple is (3,4,5).Our solution to this problem is found in Theorem2, part(iii).The results of this paper are expressed in Theorems2,3,4 and 5. Theorem2 contains five parts.Part(ii) states that there exists no Pythagorean triple which has the reciprocal property R(2,k,1). Part(iv) says that if k is at least2; then there exists no Pythagorean triple that has the reciprocal property R(1,k,l). Theorem3 states that if the positive integer v is such that the integers v-1 and v+1 are twin primes; then there is no Pythagorean triple that has the reciprocal property R(v,k,1).