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Questions about Pythagorean triples (lemmy.world)

submitted 7 months ago* (last edited 6 months ago) by Squorlple@lemmy.world to c/math@lemmy.world

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I’m gathering data for a hobby project and notating where the triangles in the data correspond to Pythagorean triples, but sometimes it doesn’t seem clear to me with certain data.

Can there exist Pythagorean triples in which the leg lengths are not coprime with each other but both are coprime with the hypotenuse? i.e., a right triangle in which (leg~1~, leg~2~, hypotenuse) = (a * n, b * n, c), in which a, b, c, and n are whole numbers and n is not a factor of c?

How can I determine if a right triangle with given lengths can scale to be a Pythagorean triple? If any of the values in (leg~1~: leg~2~: hypotenuse) are irrational, that does indeed mean the values cannot scale to be whole numbers?

Once it is determined that the triangle can scale to a Pythagorean triple, what is the best method of scaling the values to three whole numbers?

Thanks for any help

Edit: I’ve found an effective way to determine primitive Pythagorean triples from given leg lengths. Using a calculator that can output in fractional form, such as wolfram alpha, input leg~1~ / leg~2~ and the output will be a fraction with the numerator and denominator denoting the leg lengths of a primitive Pythagorean triple. Determining the hypotenuse is then simply using the Pythagorean Theorem.

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[–] ns1@feddit.uk 4 points 7 months ago

Some answers here without proof! I'm sure someone will correct me if I've got anything wrong.

Can there exist Pythagorean triples in which the leg lengths are not coprime with each other but both are coprime with the hypotenuse?

I don't think so. In your example n^2^ is a factor of c^2^ which implies n is a factor of c. Proving that is nontrivial though.

How can I determine if a right triangle with given lengths can scale to be a Pythagorean triple?

It can if the ratios between the sides are all rational. This is necessarily true if the sides are rational, but they could also all be irrational, for example 3π, 4π, 5π. It's not possible if some sides are rational and some are irrational.

Once it is determined that the triangle can scale to a Pythagorean triple, what is the best method of scaling the values to three whole numbers?

The simplest way is to first make sure it's rational and then multiply by the denominators to get integers. If the goal is to get the smallest possible integers (a primitive Pythagorean triple) you can then divide by the highest common factor.