Abstract:
Using the Euclid’s formula, we obtain an alternative formula for generating Pythagorean triples, both primitive
and non-primitive. It easy to classify Pythagorean triples using this formula based on the divisibility of the leg of a
Pythagorean triple by any positive integer. The differences in lengths between the hypotenuse and the legs of a
Pythagorean triple obtained by this alternative formula form Quadratic sequences. These quadratic sequences have
applications in various fields such as tiling
Description:
"There exist several techniques used to generate Pythagorean triples. The
most effective formula for generating Pythagorean triples is the Euclid’s formula. Whereas
the Euclid’s formula generate infinitely many Pythagorean triples, it does not generate all of
them. For instance, the Euclid’s formula generates the triple (3, 4, 5) but does not generate
(4, 3, 5), in which case a transposition is needed. In addition, the triple (9, 12, 15) cannot
be generated directly from the Euclid’s formula but rather a multiplier to the triple (3, 4, 5)
does so. In this note, we establish a formula which generates all Pythagorean triples, primitive
and non-primitive, without using a transformation and without using a multiplier."
