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Results tagged with diophantine-equations
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user 9232
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 ...
9
votes
Sum of two consecutive squares equals difference of two consecutive cubes
You can diagonalise the equation $2a(a+1)=3b(b+1)$ by writing it as $$2(2a+1)^2+1=3(2b+1)^2.$$
To parametrise all rational solutions of $2a(a+1)=3b(b+1)$ you can now homogenise $$2(2a+1)^2+1=3(2b+1) …
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