Related papers: An elementary sieve

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…

If n is a positive integer, let h(n) denote the maximal value of the product of distinct primes whose sum does not exceed n. We give some properties of this function h and describe an algorithm able to compute h(n) for large values of n.

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

This paper presents an analysis of primitive permutation groups of degree $3p$, where $p$ is a prime number, analogous to H. Wielandt's treatment of groups of degree $2p$. It is also intended as an example of the systematic use of…

We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.

In this research paper, relationship between every Mersenne prime and certain Natural numbers is explored. We begin by proving that every Mersenne prime is of the form {4n + 3,for some integer 'n'} and generalize the result to all powers of…

We develop a simple $O((\log n)^2)$ test as an extension of Proth's test for the primality for $p2^n+1$, $p>2^n$. This allows for the determination of large, non-Sierpinski primes $p$ and the smallest $n$ such that $p2^n+1$ is prime. If $p$…

In this paper we prove some new series for $1/\pi$ as well as related congruences. We also raise several new kinds of series for $1/\pi$ and present some related conjectural congruences involving representations of primes by binary…

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

We use Experimental Mathematics and Symbolic Computation (with Maple), to search for lots and lots of Perrin- and Lucas- style primality tests, and try to sort the wheat from the chaff. More impressively, we find quite a few such primality…

We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.

We show that if you represent all primes with less than n-digits as points in n-dimensional space, then they can be stored and retrieved conveniently using n-dimensional geometry. Also once you have calculated all the prime numbers less…

The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…

Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we…

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all…

For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log…

We report on the results of a computer search for primes $p$ which divide an Harmonic number $H_{\lfloor p/N \rfloor}$ with small $N > 1$.