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Results tagged with diophantine-equations
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user 8621
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
4
votes
Non-coprime solutions to x^n+y^n = z^2
The answer to the question depends on the parity of $n$.
Suppose that $n=2k$ is even. Let $(x,y,z)$ be a nonzero solution to your equation, with $y\neq 0$.
Then $(x/y,1,z/y^k)$ is also a solution of …
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