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

Two Digit Theorems

We prove that if p is a prime with a primitive root 2 then S_p(2^p)=p and give a sufficient condition for an equality of kind S_p(2^p)=+or-p.Comment: 3 page

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

Exponentially SS-numbers

Let $\mathbf{S}$ be the set of all finite or infinite increasing sequences of positive integers. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{S},$ let us call a positive number $N$ an exponentially $S$-number $(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$ Let us accept that $1\in E(S).$ We prove that, for every sequence $S\in \mathbf{S}$ with $s(1)=1,$ the exponentially $S$-numbers have a density $h=h(E(S))$ such that $$\sum_{i\leq x,\enskip i\in E(S)} 1 = h(E(S))x+R(x), where R(x) does not depend on $S$ and $h(E(S))=\prod_{p}(1+\sum_{i\geq2}\frac{u(i)-u(i-1)}{p^i}),$ where $u(n)$ is the characteristic function of $S.$Comment: 7 pages Addition three new example

On Excess of the Odious Primes

We give a more strong heuristic justification of our conjecture on the excess of the odious primes