Related papers: Differences of Harmonic Numbers and the $abc$-Conj…

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop…

Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if…

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been…

The harmonic numbers $H_n=\sum_{0<k\le n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences:…

In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. Our short paper uses a simple modification of Euler's argument to obtain new…

In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical…

A set of natural numbers will be called \emph{harmonic} if the reciprocals of its elements form a divergent series. An ultrafilter of the natural numbers will he called \emph{harmonic} if all each members are harmonic sets. The harmonic…

The monic sequence that shifts left under convolution with itself is the Catalan numbers with 130+ combinatorial interpretations. Here we establish a combinatorial interpretation for the monic sequence that shifts left under composition: it…

Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much…

We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…

We prove that, for any prime number $p\geq 5$, the set of natural numbers $n$ such that $p\mid H_n$ is finite.

Legendre's conjecture states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn <…

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

We create a simple test for distinguishing between sets of primes and random numbers using just the sum-of-digits function. We find that the sum-of-the-digits of prime numbers does not have an equal probability of being odd or even. The…

In 2013, Joerg Arndt recorded that the Fibonacci numbers count integer compositions where the first part is greater than the second, the third part is greater than the fourth, etc. We provide a new combinatorial proof that verifies his…

Let $k$ and $n$ be natural numbers. Let $\omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of…

Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$,…

We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly,…

Let $d\ge4$ and $c\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\le d\le 36$), the smallest prime $p\equiv c\pmod d$ with $p\ge(2dn-c)/(d-1)$ is the…

During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously…