Related papers: Around Wilson's theorem

We call positive integer n a near-perfect number, if it is sum of all its proper divisors, except of one of them ("redundant divisor"). We prove an Euclid-like theorem for near-perfect numbers and obtain some other results for them.

Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis,…

Let $x$ be a real number satisfying $x \geq 2$. For any positive integer $n$, we define $s(n)$ as the smallest non-negative integer such that $n + s(n)$ is a perfect square. In this paper, we derive an asymptotic formula for the sum…

We obtain an estimate for the cubic Weyl sum which improves the bound obtained from Weyl differencing for short ranges of summation. In particular, we show that for any $\varepsilon>0$ there exists some $\delta>0$ such that for any coprime…

For $n,\,d\ge1$ let $p(n,2d)$ denote the smallest number $p$ such that every sum of squares of forms of degree $d$ in $\mathbb{R}[x_1,\dots,x_n]$ is a sum of $p$ squares. We establish lower bounds for these numbers that are considerably…

Using product integrals we review the unambiguous mathematical representation of Wilson line and Wilson loop operators, including their behavior under gauge transformations and the non-abelian Stokes theorem. Interesting consistency…

The classical theorem of Birkhoff states that the $T^N f(x) = (1/N)\sum_{k=0}^{N-1} f(\sigma^k x)$ converges almost everywhere for $x\in X$ and $f\in L^{1}(X)$, where $\sigma$ is a measure preserving transformation of a probability measure…

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a…

We start with an n-point correlation function in a conformal gauge theory. We show that a special limit produces a polygonal Wilson loop with $n$ sides. The limit takes the $n$ points towards the vertices of a null polygonal Wilson loop…

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say $I_{\Lambda}(N,h)$, of the von Mangoldt function $\Lambda(n)$ ($:= \log p$ for prime-powers $n=p^r$, 0…

Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce…

For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.

Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i<j\leq…

Suppose $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\leq 1$, meaning $|x_i - x_j| \leq 1$ for all $1 \leq i,j \leq n$. We show that the ratio of the number of ``neighbors''…

The Euler product formula relates Dirichlet $L(s,\chi)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re…

We prove that every sufficiently large integer $n$ can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every $n > 24$ and prove two results to support this claim. First,…

We show that, for almost all $x$, the interval $(x, x+(\log x)^{2.1}]$ contains products of exactly two primes. This improves on a work of the second author that had $3.51$ in place of $2.1$. To obtain this improvement, we prove a new type…

For a bounded measurable set $A\subseteq \mathbb{R}$ we denote the Lebesgue measure of $\{(x, y)\in A^2\colon x\le y\le x+1\}$ by $\Phi(A)$. We prove that if $I=A_1\cup\dots\cup A_{k+1}$ partitions an interval $I$ of length $L$ into $k+1$…