Related papers: A Dynamical Proof of the Prime Number Theorem
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…
We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…
New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…
We present a probabilistic proof of Euler's pentagonal number theorem based on a shuffling model.
We present in this work a new and simple proof of the false centre theorem.
By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical…
We illustrate the concept of mathematical proof.
We introduce an elementary argument to the theory of distribution of sequences modulo one.
The goal of this expository article is to present a proof that is as direct and elementary as possible of the fundamental theorem of complex multiplication (Shimura, Taniyama, Langlands, Tate, Deligne et al.). The article is a revision of…
We study Kummer's approach towards proving the Fermat's last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other…
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
As a corollary to the recent extraordinary theorem of Maynard and Tao, we re-prove, in a stronger form, a result of Shiu concerning "strings" of consecutive, congruent primes.
Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins
We give a simple Tauberian proof of the Prime Number Theorem using only elementary real analysis. Hence, the analytic continuation of Riemann's zeta function $\zeta$ and its non-vanishing value on the whole line $\{z\in {\mathbb…
Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$. We also show, as a corollary of the Green-Tao theorem, that there are arbitrarily long…
Simple divisibility rules are given for the 1st 1000 prime numbers.
We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.