Three theorems on twin primes

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a very simple conjectural algorithm of constructing a pair $(p,\enskip p+2)$ of twin primes over arbitrary given integer $m\geq4$ such that $p+2\geq m.$Comment: 17 pages. New section: "A theorem on twin primes which is independent on observation of type 6)

Theorems on twin primes-dual case

We prove dual theorems to theorems proved by author in \cite {5}. Beginning with Section 10, we introduce and study so-called "twin numbers of the second kind" and a postulate for them. We give two proofs of the infinity of these numbers and a sufficient condition for truth of the postulate; also we pose several other conjectures. Finally, we consider a conception of axiom of type "AiB".Comment: 26 pages. Correction of Remark 6 arXiv admin note: text overlap with arXiv:0911.547

A Conjecture on Primes and a Step towards Justification

We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.Comment: 14page

On Monotonic Strengthening of Newman-like Phenomenon on (2m+1)-multiples in Base 2m

We obtain exact and asymptotic expressions for the excess of nonnegative (2m+1)-multiples less than (2m)^k with even digit sums in the base 2m.Comment: 5 page

On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2

We obtain a simple relations for the Newman sum over multiples of a prime with a primitive or semiprimitive root 2. We consider the case of p=17 as well.Comment: 4 page

A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function σx(n)\sigma_x(n)

For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove a recursion for divisor function $\sigma_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ As a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function $\sigma_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$Comment: 11 pages, improvement of the text of Introduction; addition of Section

On Stephan's conjectures concerning Pascal triangle modulo 2

We prove a series of Stephan's conjectures concerning Pascal triangle modulo 2 and give a polynomial generalization.Comment: Adding two reference

On Unique Additive Representations of Positive Integers and Some Close Problems

Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion relation for Moser's numbers. We also study interesting properties of the sequence (rm_r(n-1)+1),n>=1, and its connection with some important problems. In particular, in the case of r=2 this sequence is surprisingly connected with the numbers solving the combinatorial Josephus-Groer problem. We pose also some open questions.Comment: 14 pages; removing of the last section (Section 10) in view I found an error in proof. in proo

Process of "Primoverization" of Numbers of the Form a^n-1

We call an integer N>1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of numbers of the form a^n-1.Comment: 6 pages; 4 additional theorem

On Erd\H{o}s constant

In 1944, P. Erd\H{o}s \cite{1} proved that if $n$ is a large highly composite number (HCN) and $n_1$ is the next HCN, then $$n<n_1<n+n(\log n)^{-c},$$ where $c>0$ is a constant. In this paper, using numerical results by D. A. Corneth, we show that most likely $c<1.$Comment: 3 page