Related papers: Folkman's theorem and the primes
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…
Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…
We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.
Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.
In this short note we show how the higher index theory can be used to prove results concerning the non-existence of complete riemannian metric with uniformly positive scalar curvature at infinity. By improving some classical results due to…
We start by presenting a theory of finite sets using the approach which is essentially that taken by Whitehead and Russell in Principia Mathematica}, and which does not involve the natural numbers (or any other infinite set). This theory is…
Ova-angular rotations of a prime number are characterized, constructed using the Dirichlet theorem. The geometric properties arising from this theory are analyzed and some applications are presented, including Goldbach's conjecture, the…
The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent…
Number of results in number theory have been developed using a new method. The Goldbach binary conjecture in strengthened formulation have been among them.
We develop infinite-dimensional Ramsey theory for Fra\"iss\'e limits of finitely constrained free amalgamation classes in finite binary languages. We show that our approach is optimal and in particular, recovers the exact big Ramsey degrees…
In Wilson's Theorem the primality of a number hinges on a congruence. We present a similar test where the primality of a number m hinges, instead, on the indivisibility of 4(m-5)! by m. One implication of this theorem is a necessary and…
The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.
We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cram\'{e}r's, Shanks', and five Smarandache's conjectures. We make use of both…
Hirst investigated a slight variant of Hindman's Finite Sums Theorem -- called Hilbert's Theorem -- and proved it equivalent over $\RCA_0$ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural…
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…
We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are…
Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has…
The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre…
We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.
By operations on models we show how to relate completeness with respect to permissive-nominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal…