In this note we prove that the equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1=q^n+1$, $q\ge 2,n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k{10}^{{10}^{182}}$, $q{10}^{{10}^{165}}$ and $n2·{10}^{17}$.

Let ${𝔖}_n$ denote the symmetric group with $n$ letters, and $g\left(n\right)$ the maximal order of an element of ${𝔖}_n$. If the standard factorization of $M$ into primes is $M=q_1^{{\alpha }_1}{q}_2^{{\alpha }_2}...q_k^{{\alpha }_k}$, we define $\ell \left(M\right)$ to be $q_1^{{\alpha }_1}+q_2^{{\alpha }_2}+...+q_k^{{\alpha }_k}$; one century ago, E. Landau proved that $g\left(n\right)={max}_{\ell \left(M\right)\le n}M$ and that, when $n$ goes to infinity, $logg\left(n\right)\sim \sqrt{nlog\left(n\right)}$. There exists a basic algorithm to compute $g\left(n\right)$ for $1\le n\le N$; its running time is $𝒪\left(N^{3/2}/\sqrt{logN}\right)$ and the needed memory is $𝒪\left(N\right)$; it allows computing $g\left(n\right)$ up to, say, one million. We describe an algorithm to calculate $g\left(n\right)$ for $n$ up to ${10}^{15}$. The main idea is to use the...

We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$ $(s=1,2,\cdots )$ by rationals $x/y$, such that $x,y$ satisfy a quadratic equation. For instance, all positive integers $x,y$ with $y\equiv 0(mod2)$ solving the Pythagorean equation $x^2+y^2=z^2$ satisfy $$|ysinh(1/s)-x|\gg \frac{loglogy}{logy}.$$ Conversely, for every $s=1,2,\cdots$ there are infinitely many coprime integers $x,y$, such that $$|ysinh(1/s)-x|\ll \frac{loglogy}{logy}$$ and $x^2+y^2=z^2$ hold simultaneously for some integer $z$. A generalization to the approximation of $h\left(e^{1/s}\right)$ for rational...

For every positive integer $n$ let $p\left(n\right)$ be the largest prime number $p\le n$. Given a positive integer $n=n_1$, we study the positive integer $r=R\left(n\right)$ such that if we define recursively $n_{i+1}=n_i-p\left(n_i\right)$ for $i\ge 1$, then $n_r$ is a prime or $1$. We obtain upper bounds for $R\left(n\right)$ as well as an estimate for the set of $n$ whose $R\left(n\right)$ takes on a fixed value $k$.

We provide upper bounds for the mean square integral $${\int }_X^{2X}{\left({𝔻}_k(x+h)-{𝔻}_k\left(x\right)\right)}^2\mathrm{d}x,$$ where $h=h\left(X\right)\gg 1,h=o\left(x\right)\mathrm{as}X\to \infty$ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, ${𝔻}_k\left(x\right)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k\left(n\right)$, generated by ${\zeta }^k\left(s\right)$.