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Results tagged with diophantine-equations
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user 37555
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 ...
14
votes
Perfect powers in the solutions of a certain Pell equation
The standard appproach is via Baker's method of linear forms in logarithms. We have $x_n+\sqrt{3}y_n=(2+\sqrt{3})^n$, thus $2x_n=(2+\sqrt{3})^n+(2-\sqrt{3})^n$. Now assume that $x_n=7^m$, and consider …
4
votes
How to prove that this equation has only one solution?
Unless $3^P$ is very close to $2^{P+Q}$, the right hand side will be smaller than 1. Hence the linear form $(P+Q)\log 2 - P\log 3$ is exceptionally small, and you should be able to obtain effective up …
4
votes
Failing of heuristics from circle method
Note as a caveat that the heuristics can fail in the other direction as well: Dietmann and Elsholtz ( http://www.math.tugraz.at/~elsholtz/WWW/papers/papers26de08.pdf ) have shown that if $p\equiv 7\pm …
3
votes
Bounds on near-zero integer linear combinations of numbers linearly independent over $\mathb...
There is a lower bound $f(N)>N^{N-101}$. To see this suppose without loss that $\alpha_1=1$. Then consider all the $N^{N-1}$ values of $\sum_{i=2}^{N} m_i\alpha_i\bmod 1$ with $|m_i|\leq N/2$. There i …