Related papers: Some remarks on regular integers modulo $n$

We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,\phi(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$,…

We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition…

Fix an integer $h \geq 2$, and let $b_1, \ldots, b_h$ be (not necessarily distinct) positive integers with $\gcd(b_1, \ldots, b_h) = 1$. For any subset $A \subseteq \mathbb{N}$, let $r_A(n)$ denote the number of solutions $(k_1, \ldots,…

Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is…

Let $k_i\in \mathbb N$ $(i\ge 1)$ satisfy $2\le k_1\le k_2\le \ldots $. Freiman's theorem shows that when $j\in \mathbb N$, there exists $s=s(j)\in \mathbb N$ such that all large integers $n$ are represented in the form…

Let $a$ and $m>0$ be integers. We show that for any integer $b$ relatively prime to $m$, the set $\{a^n+bn:\ n=1,\ldots,m^2\}$ contains a complete system of residues modulo $m$. We also pose several conjectures for further research; for…

For a fixed rational number g different from -1,0,1 and integers a and d the set N_g(a,d) of primes p for which the order of g(mod p) is congruent to a(mod d) is considered. It is shown, assuming the Generalized Riemann Hypothesis (GRH),…

The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study…

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta <…

Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is…

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…

Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the…

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class…

We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the…

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime…

Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a pair of conjugate…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = \nu_{q}(N) = 1$ to hold. We also prove that $q^k < 2/3{n^2}$, and that $I(q^k) < I(n)$,…

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations…

We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.