Related papers: A Dynamical Proof of the Prime Number Theorem
This work is meant to demonstrate new class of prime numbers -- cyclic prime numbers, that can be derived from any prime number at certain numeric systems. Cyclic prime numbers are also related to the cyclic numbers and full reptend prime…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
An technically interesting proof of a known theorem.
Inspired by a recent preprint of N. Curien, we provided what may be a new and elementary proof of the Law of Large Numbers.
It is shown that the Mean Value Theorem for arithmetic functions, and simple properties of the zeta function are sufficient to assemble proofs of the Prime Number Theorem, and Dirichlet Theorem. These are among the simplest proofs of the…
In this note we prove an inequality involving primes and the product of consecutive primes.
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.
This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger…
This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration,…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
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…
The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.
We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by…
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…
In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…
Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…
Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…