User:Lorenzo Sauras Altuzarra

Website: lorenzosaurasaltuzarra.com
A problem on the geometry of numbers: if $n$ $n$ and $u$ $u$ are integers such that $n>1$ $n>1$ and $|u|=1$ $|u|=1$ , and $a$ $a$ (resp., $v$ $v$ ) is a point of $\mathbb {Z} ^{n}$ $\mathbb {Z} ^{n}$ whose components exceed one (resp., have modulus one), when does the set of points $p$ $p$ of $\mathbb {Q} ^{n}$ $\mathbb {Q} ^{n}$ for which $a_{1}a_{2}\ldots a_{n}+u$ $a_{1}a_{2}\ldots a_{n}+u$ divides $v_{1}a_{1}^{p_{1}}+v_{2}a_{2}^{p_{2}}+\ldots +v_{n}a_{n}^{p_{n}}$ $v_{1}a_{1}^{p_{1}}+v_{2}a_{2}^{p_{2}}+\ldots +v_{n}a_{n}^{p_{n}}$ equal the first orthant of some shifted point-lattice?
- Note I: given any positive integer $c$ , the Diophantine equation $a_{1}^{x_{1}}-a_{2}^{x_{2}}=c$ is a Pillai equation and the Diophantine equation $v_{1}a_{1}^{x_{1}}+v_{2}a_{2}^{x_{2}}=c$ is a subkind of generalized Pillai equation.
- Note II: the affirmative form of the case in which $n=2$ and $u=v_{1}=v_{2}=1$ is Conjecture 4 from my article "Some applications of Baaz’s generalization method to the study of the factors of Fermat numbers".

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