97 research outputs found

    Three theorems on twin primes

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    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,p+2)(p,\enskip p+2) of twin primes over arbitrary given integer mβ‰₯4m\geq4 such that p+2β‰₯m.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

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    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

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

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    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

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    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

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    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

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    In 1944, P. Erd\H{o}s \cite{1} proved that if nn is a large highly composite number (HCN) and n1n_1 is the next HCN, then n<n1<n+n(log⁑n)βˆ’c,n<n_1<n+n(\log n)^{-c}, where c>0c>0 is a constant. In this paper, using numerical results by D. A. Corneth, we show that most likely c<1.c<1.Comment: 3 page

    Exponentially SS-numbers

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    Let S\mathbf{S} be the set of all finite or infinite increasing sequences of positive integers. For a sequence S={s(n)},nβ‰₯1,S=\{s(n)\}, n\geq1, from S,\mathbf{S}, let us call a positive number NN an exponentially SS-number (N∈E(S)),(N\in E(S)), if all exponents in its prime power factorization are in S.S. Let us accept that 1∈E(S).1\in E(S). We prove that, for every sequence S∈SS\in \mathbf{S} with s(1)=1,s(1)=1, the exponentially SS-numbers have a density h=h(E(S))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 SS and h(E(S))=∏p(1+βˆ‘iβ‰₯2u(i)βˆ’u(iβˆ’1)pi),h(E(S))=\prod_{p}(1+\sum_{i\geq2}\frac{u(i)-u(i-1)}{p^i}), where u(n)u(n) is the characteristic function of S.S.Comment: 7 pages Addition three new example

    On Excess of the Odious Primes

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

    Equations of the form t(x+a)=t(x)t(x+a)=t(x) and t(x+a)=1βˆ’t(x)t(x+a)=1-t(x) for Thue-Morse sequence

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    For every aβ‰₯1a\geq1 we give a recursion algorithm of building of set of solutions of equations of the form t(x+a)=t(x)t(x+a)=t(x) and t(x+a)=1βˆ’t(x),t(x+a)=1-t(x), where {t(n)}\{t(n)\} is Thue-Morse sequence. We pose an open problem and two conjectures.Comment: 14 pages. Adding in proof of Theorem 2 in detail point
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