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Joined 1 year ago
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Cake day: November 7th, 2024

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  • I try to answer your question with an example, maybe I misunderstood.

    16 + 9 = 25
    4*4 + 3*3 = 5*5
    

    4, 3 and 5 are prime numbers, and form the Pythagorean triple (3, 4, 5).

    Both legs 4 and 3 are coprime: gcd(3, 4) = 1. and both are coprime with the hypotenuse gcd(3, 5)=1 and gcd(4, 5) = 1. Now lets multiply this triple by 7. Obviously a new Pythagorean triple is formed (7*3, 7*4, 7*5).

    But coprime-ness of the legs is lost because of the multiplication, this is not a counter example to your theory:

    1. Leg lenghts are not co-prime with each other ✅ gcd(7*3, 7*4) = 7 > 1
    2. Leg lengths are coprime with the hypotenuse ☠ No.

    I try to translate this example to a proof.

    Given a Pythagorean triple (a, b, c) where a, b and c are integers that fullfill the equation (I) a*a+b*b = c*c. Further say a and b share a greatest common factor x>1. This means gcd(a, b) = x. This means a and b are not coprime, which is your first constraint.

    This means we can write a as x*u and b as x*v, where u and v are integers. Now place this in equation (I)

    (x*u)*(x*u) + (x*v)*(x*v) = c*c
    ↔ x*x*u*u + x*x*v*v = c*c | commutative and associative law
    ↔ x*x (u*u+v*v) = c*c
    

    Since x, u, v and c are integers, this means c has to share the common factor x with a and b. Which means gcd(a, c) >= x, which is a contradiction with gcd(a, c) = gcd(b, c) = 1 (legs are co-prime with hypotenuse).

    Can someone write this proof better please lol, and pls correct if wrong.















  • The guy has pretty big constraints on his setup: Voice Cloning AND open source …

    At the top of the leaderboard is Kokoro. Kokoro is an amazing model! Especially for a modest 82 Million (!) parameters and a mere 360 MB (!). However, like many models in this leaderboard - I can’t use it - since it doesn’t support voice cloning.