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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.
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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 $pmathbb{Z}_p[zeta_{p^2-1}]$ with $gequiv zeta_{p^2-1}pmod {pmathbb{Z}_p[zeta_{p^2-1}]}$. Let $Deltaequiv3pmod4$ be an arbitrary quadratic non-residue modulo $p$ in $mathbb{Z}$. By the Local Existence Theorem we know that $mathbb{Q}_p(sqrt{Delta})=mathbb{Q}_p(zeta_{p^2-1})$. For all $xinmathbb{Z}[sqrt{Delta}]$ and $yinmathbb{Z}_p[zeta_{p^2-1}]$ we use $bar{x}$ and $bar{y}$ to denote the elements $xmod pmathbb{Z}[sqrt{Delta}]$ and $ymod pmathbb{Z}_p[zeta_{p^2-1}]$ respectively. If we set $a_k=k+sqrt{Delta}$ for $0le kle p-1$, then we can view the sequence $$S := overline{a_0^2}, cdots, overline{a_0^2n^2}, cdots,overline{a_{p-1}^2}, cdots, overline{a_{p-1}^2n^2}cdots, overline{1^2}, cdots,overline{n^2}$$ as a permutation $sigma$ of the sequence $$S^* := overline{g^2}, overline{g^4}, cdots,overline{g^{p^2-1}}.$$ We determine the sign of $sigma$ completely in this paper.
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 are a new Type II estimate and using this estimate by applying Harmans sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selbergs eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
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 proofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
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 features both with 3d bosonization and with 4d Seiberg duality. We provide a few consistency checks of the proposal, mapping the structure of vacua and performing perturbative computations in the $varepsilon$-expansion.
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-electron second-order effects in $alpha$. When compared to approximate QED computations, a notable discrepancy is found especially for F-like uranium for which the predicted self-energy contributions even differ in sign. Moreover, all deviations between theory and experiment for the $^2P_{1/2}$ - $^2P_{3/2}$ fine-structure energies of F-like ions, reported recently by Li et al., Phys. Rev. A 98, 020502(R) (2018), are resolved if their highly accurate, non-QED fine-structure values are combined with the QED corrections ab initially evaluated here.
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