Related papers: On digits of Mersenne numbers

We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent…

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $\psi(x;q,a)$ and $\psi(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for…

The purpose of this note is to report on the discovery of the primes of the form $p=1+n!\sum n$, for some natural numbers $n>0$. The number of digits in the prime p are approximately equal to $\lfloor log_{10}(1+n!\sum n)\rceil+1$.

For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for…

This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n \pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d., and $a_n$ and $b_n$ are independent with $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and…

A classical problem in analytic number theory is to study the distribution of fractional part $\alpha p+\beta$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes…

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the…

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

Let $p$ be a prime $\equiv 3$ mod 4, $p>3$, and suppose that 10 has the order $(p-1)/2$ mod p. Then $1/p$ has a decimal period of length $(p-1)/2$. We express the frequency of each digit $0,\ldots,9$ in this period in terms of the class…

A classical problem in analytic number theory is to study the distribution of $\alpha p$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is…

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$,…

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even…

An odd prime $p$ is called irregular with respect to Euler polynomials if it divides the numerator of one of the numbers $$E_1(0),E_{3}(0),\ldots,E_{p-2}(0),$$ where $E_n(x)$ is the $n$-th Euler polynomial. As in the classical case, we link…

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1, n_2, n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split…

We study number theoretic properties of the map $x \mapsto x^{x} \mod{p}$, where $x \in \{1,2,\ldots,p-1\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map…

The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{2})$ is studied in detail with a focus on representing Mersenne primes in the form $x^{2}+7y^{2}$. It…

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar…

We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result:…