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Results tagged with diophantine-equations
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user 177744
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 ...
6
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3
answers
593
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Non-trivial solution to $\sum^{n}_{i=1}\sum^{n}_{j=1,j\ne i}(x_{i})^{(x_j)}=(\sum^{n}_{i=1}x...
This problem was first asked at Mathematics Stack Exchange, where it wasn't drawn much attention.
For ease of reading,
$$S=\sum_{i=1}^nx_i, S_p=\sum_{i=1,i\ne p}^nx_i, S^{[q]}=\sum_{i=1}^nx_i^q, S_p^{ …
0
votes
Non-trivial solution to $\sum^{n}_{i=1}\sum^{n}_{j=1,j\ne i}(x_{i})^{(x_j)}=(\sum^{n}_{i=1}x...
The results are shown in this answer.
The upper bound of $S^{S_n}$.
I proved that $S^{S_n}\le n(n-1)$.
@mathlove commented that $S^{S_n}\le n(n-1)/2$.
I proved that $S^{S_n}\le\frac{n^2-3n-6}2$.
The …