On a Generalised Lehmer Problem for Arbitrary Powers

关于模素数广义Lehmer数存在性的研究

On existence of generalised Lehmer numbers modulo primes

本文推广Lehmer数为模奇素数$q$的$q$-元组，计数并给出带显式误差项的渐近估计，利用特征和与Deligne界。

A Lucas-Lehmer approach to generalised Lebesgue-Ramanujan-Nagell equations

We describe a computationally efficient approach to resolving equations of the form $C_1x^2 + C_2 = y^n$ in coprime integers, for fixed values of $C_1$, $C_2$ subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

Lehmer's Problem for arbitrary groups

We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.

The generalised Baker--Schmidt problem on hypersurfaces

The Generalised Baker--Schmidt Problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $ψ$-approximable points on a nondegenerate manifold. There are two variants of this problem, concerning simultaneous and dual approximation. Beresnevich--Dickinson--Velani (in 2006, for the homogeneous setting) and Badziahin--Beresnevich--Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The corresponding convergence counterpart represents a major challenging open question and the progres…

Exact & Numerical Tests of Generalised Root Identities for non-integer μ

We consider the generalised root identities introduced in [1] for simple functions, and also for Γ(z+1) and ζ(s). In this paper, unlike [1], we focus on the case of noninteger μ. For the simplest function f(z)=z, and hence for arbitrary polynomials, we show that they are satisfied for arbitrary real μ (and hence for arbitrary complex μ by analytic continuation). Using this, we then develop an asymptotic formula for the derivative side of the root identities for Γ(z+1) at arbitrary real μ, from which we are able to demonstrate numerically that Γ(z+1) also satisfies the generalised root identiti…

On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.