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"Special Case of Fermat's Last Theorem" (mentor: Shin Eui Song) UMD Mathematics Directed Reading Program

Suppose that we want to classify all Primitive Pythagorean Triples, solutions of the equation x^2 + y^2 = z^2 with gcd(x,y,z)= 1. We can factor the equation as (x+yi)(x-yi)=z^2. Using more theory, one can show that x+yi has the form of uα^2, for some unit u and element α in the gaussian integers. The only units are ±1 and ±i, and set α = m+ni. Then using that, we can classify all primitive Pythagorean triples as {x,y}={±(m^2 - n^2), ±2mn}, with z=±(m^2 + n^2), with all combinations of signs allowed. Here they are mapped as coordinates, based on every choice of m and n.