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MIT PRIMES/Art of Problem Solving

CROWDMATH 2018: Neural Codes

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A corrolary to the abc-conjecture
SCP   1
N Nov 7, 2018 by JGeneson
This is just a fun fact taken from my master thesis, related to Problem $4$ which is formulated very rough (and so this is not giving "the" solution).

Analogously to the original congruent numbers (see e.g. https://en.wikipedia.org/wiki/Congruent_number), one can define a 3D version, being the cubefree part of the volume of a right triangular pyramid with (6) integer side lengths.
In the 2D case, one can get an infinite family (even all) integer solutions by Pythagorean triples.
Similarly, in the 3D case, one has the infinite family generated by Sounderson's formula (see generating formula at https://en.wikipedia.org/wiki/Euler_brick )

In the normal setting, any squarefree number which is congruent, is the area of infinitely many primitive right triangles with integer side lengths.
As a corrolary of the abc-conjecture, this is not the case for 3D congruent numbers when restricting to the solutions generated by Sounderson's formula.



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SCP
Sep 12, 2018
JGeneson
Nov 7, 2018
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