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Results tagged with diophantine-equations
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user 35416
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 ...
17
votes
The modular arithmetic contradiction trick for Diophantine equations
For diagonal conics, such as your example of $x^2+y^2-3z^2=0$, a non-zero rational solution exists if and only if one exists modulo all powers of all prime divisors of the coefficients and modulo powe …
- 13k
13
votes
Fermat's last theorem over larger fields
There might well be an elementary construction of infinitely many points (which I cannot think of right now), but in any case, I think that there are experts out there who expect there to be infinitel …
- 13k
9
votes
Accepted
A remark of Mordell alluding to a local/global principle for cubic Diophantine equations
As Franz says, Mordell is talking about the conjecture of Birch and Swinnerton-Dyer. But I just wanted to add that in the modern formulation of the conjecture, it is not easy to discern the original h …
- 13k