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Vladimir Shevelev

Publications and source records attributed to Vladimir Shevelev.

At least 55 records · Page 3Linked to original sources
arXiv2009

Very small intervals containing at least three primes

Let $p_n$ is the $n$-th prime. With help of the Cramér-like model, we prove that the set of intervals of the form $(2p_n,\enskip2p_{n+1})$ containing at list 3 primes has a positive density with respect to the set of all intervals of such form.

math.NT↗
arXiv2009

Gold ratio and a trigonometric identity

We give two proofs of the identity $$\sqrt{\frac{\cos\frac{2π}{5}} {\cos\fracπ{5}}}+\sqrt{\frac{\cos\fracπ{5}} {\cos\frac{2π}{5}} }=\sqrt{5},$$ using and not using the gold ratio.

math.HO↗
arXiv2009

Binomial predictors

For a prime p and nonnegative integers n,k, consider the set A_{n,k}^{(p)}={x is in [0,1,...,n]: p^k||binom {n} {x}}. Let the expansion of n+1 in base p be: n+1=alpha_{0} p^ν+alpha_{1}p^{nu-1}+...+alpha_{nu}, where 0<=alpha_{i}<= p-1,i=0,...,nu. Then the number n is called a binomial predictor in base p,if |A_{n,k}^{(p)}|=alpha_{k}p^{nu-k},k=0,1,...,nu. We give a full description of the binomial predictors in base p.

math.NT↗
arXiv2009

Several results on sequences which are similar to the positive integers

Sequence of positive integers $\{x_n\}_{n\geq1}$ is called similar to $\mathbb {N}$ respectively a given property $A$ if for every $n\geq1$ the numbers $x_n$ and $n$ are in the same class of equivalence respectively $A\enskip(x_n\sim n (prop \enskip A).$ If $x_1=a(>1)\sim1 (prop\enskip A)$ and $x_n>x_{n-1}$ with the condition that $x_n$ is the nearest to $x_{n-1}$ number such that $x_n\sim n (prop \enskip A),$ then the sequence $\{x_n\}$ is called minimal recursive with the first term $a\enskip(\{x_n^{(a)}\}).$ We study two cases: $A=A_1$ is the value of exponent of the highest power of 2 dividing an integer and $A=A_2$ is the parity of the number of ones in the binary expansion of an integer. In the first case we prove that, for sufficiently large $n, \enskip x_n^{(a)}=x_n^{(3)};$ in the second case we prove that, for $a>4$ and sufficiently large $n,\enskip x_n^{(a)}=x_n^{(4)}.$

math.NT↗
arXiv2009

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 $σ_x(n)$

For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove a recursion for divisor function $σ_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ 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 $σ_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$

math.NT↗
arXiv2008

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.

math.NT↗
arXiv2008

Estimates of Newman Sum over Numbers Multiple of a Fixed Integer

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to infinity.We also discuss a connection of our results with a digit conjecture on primes.

math.NT↗
arXiv2008

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.

math.NT↗
arXiv2007

On Ramanujan Cubic Polynomials

A polynomial x^3+px^2+qx+r with the condition pr^(1/3)+ 3r^(2/3)+q=0 we call a Ramanujan cubic polynomial (RCP). We study different interest properties of RCP, in particular, an important role of a parameter pq/r. We prove some new beautiful identities containing sums of 6 cubic radicals of values of trigonometrical functions as well.

math.AC↗
arXiv2007

New Digit Results and Several Problems

We give some new relations for Newman digit sums respectively different modulos and put some problems. In particular, for the odd prime modulos we put an important conjecture.

math.NT↗
arXiv2007

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.

math.NT↗