Related papers: On Excess of the Odious Primes

We give a new proof of the existence of designs, which is much shorter and gives better bounds.

We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be…

Six conjectures on pairs of consecutive primes are listed in this paper, together with examples for each case.

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.

We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.

We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of…

Starting with Zhang's theorem on the infinitude of prime doubles, we give an inductive argument that there exists an infinite number of prime $k$-tuples for at least one admissible set $\mathcal{H}_k=\{h_1,\ldots,h_k\}$ for each $k$.

We give a visually appealing counterexample to the proposition that unbiased estimators are better than biased estimators.

We prove a conjecture of Heath-Brown on the number of rational points of bounded height for a large class of projective varieties.

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

We survey recent developments on the Restriction conjecture.

We provide empirical evidence for the Erd\H{o}s-Straus conjecture by improving computational bounds to $10^{18}$ and by evaluating the solution-counting function $f(p)$ for this conjecture.

Recently, Moret\'o and Rizo proposed a conjecture, known as the Picky Conjecture, proposing new character correspondences extending the McKay Conjecture. We prove the Picky Conjecture for all quasi-simple groups of Lie type for non-defining…

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both…

In this paper we propose an extension of Defeasible Logic to represent and compute three concepts of defeasible permission. In particular, we discuss different types of explicit permissive norms that work as exceptions to opposite…

We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least.…

The aim of this note is a proof of a recent conjecture of Kellner concerning the number of distinct prime factors of a particular product of primes. The proof uses profound results from analytic number theory, such as Granville-Ramar\'{e}'s…

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

The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic…