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In this paper, we determine all the squares in the sequence ${prod_{k=2}^n(k^2-1)}_{n=2}^infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.
Let $p=2n+1$ be an odd prime, and let $zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $overline{mathbb{Q}_p}$ of $mathbb{Q}_p$. We let $ginmathbb{Z}_p[zeta_{p^2-1}]$ be a primitive root modulo $pmathbb{Z}_p[zeta_{p^2
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, ngeq 2, xyz eq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the p
We consider minimally supersymmetric QCD in 2+1 dimensions, with Chern-Simons and superpotential interactions. We propose an infrared $SU(N) leftrightarrow U(k)$ duality involving gauge-singlet fields on one of the two sides. It shares qualitative fe
Ab initio calculations of QED radiative corrections to the $^2P_{1/2}$ - $^2P_{3/2}$ fine-structure transition energy are performed for selected F-like ions. These calculations are nonperturbative in $alpha Z$ and include all first-order and many-ele