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Assume a polynomial-time algorithm for factoring integers, Conjecture~ref{conj}, $dgeq 3,$ and $q$ and $p$ are prime numbers, where $pleq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $log(q)$ that lifts every $mathbb{Z}/qmathbb{Z}$ point of $S^{d-2}subset S^{d}$ to a $mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $mathbb{Z}/qmathbb{Z}$ points of $S^{d-2}subset S^d$.
In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $ngeq 2$ and $d=p^b$, $p$ a prime and $pleq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences
This paper concerns the behavior of the eigenfunctions and eigenvalues of the round spheres Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of particular intere
A two-type version of the frog model on $mathbb{Z}^d$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_i$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random numbe
In this paper we propose a different (and equivalent) norm on $S^{2} ({mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({mathbb{D}})$ in this norm admits an explicit form
We consider an $O(d,d;mathbb{Z})$ invariant massive deformation of double field theory at the level of free theory. We study Kaluza-Klein reduction on $R^{1,n-1} times T^{d}$ and derive the diagonalized second order action for each helicity mode. Imp