Related papers: Infinite Primes From Integer Partitions

Ferrers graphs and tables of partitions are treated as vectors. Matrix operations are used for simple proofs of identities concerning partitions. Interpreting partitions as vectors gives a possibility to generalize partitions on negative…

In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…

This short paper gives another proof of the infinitude of primes by using upper box dimension, which is one of fractal dimensions.

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a…

A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}[i]$, there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact…

We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.

In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…

We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor n\alpha\rfloor\}$, where $\alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some…

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

The symbolic representation of a number should be considered as a data structure, and the choice of data structure depends on the arithmetic operations that are to be performed. Numbers are almost universally represented using position…

The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…

Tabling in logic programming has been used to eliminate redundant computation and also to stop infinite loop. In this paper we investigate another possibility of tabling, i.e. to compute an infinite sum of probabilities for probabilistic…

The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly…