Related papers: Distribution properties of compressing sequences d…

We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which…

The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp…

Let $p\geq3$ be a prime and $e\geq2$ an integer. Let $\sigma(x)$ be a primitive polynomial of degree $n$ over $Z/p^eZ$ and $G'(\sigma(x),p^e)$ the set of primitive linear recurring sequences generated by $\sigma(x)$. A compressing map…

In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the…

We introduce a sequence representation of a random variable $X$ supported on a compact interval $[a,b]$, which we call a primitive sequence. We construct this sequence by repeatedly antidifferentiating the associated cumulative distribution…

We introduce an elementary argument to the theory of distribution of sequences modulo one.

Let $(p_n)$ denote the sequence of prime numbers, with $2=p_1<p_2<\ldots$. We demonstrate the existence of an irrational number $\alpha$ having the property that the sequence $(\alpha p_n)$ is not well-distributed modulo $1$.

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…

We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.

Let b be an odd integer such that b=+/-1 (mod 8) and let q be a prime with primitive root 2 such that q does not divide b. We show that if (p(k)) is a sequence of odd primes, with 0<=k<=q-2 such that p(k)=2p(k-1)+b for all 1<=k<=q-2, then…

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…

We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the…

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other…

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…

We study types that appear in ultraproducts that have distributions which can be thought of as a sequence of graphs. The property of having distributions that are captured by graphs is motivated by a commonality of $\mathrm{SOP}_2$-types…

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

Linear second order recursive sequences with arbitrary initial conditions are studied. For sequences with the same parameters a ring and a group is attached, and isomorphisms and homomorphisms are established for related parameters. In the…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

Compression of integer sets and sequences has been extensively studied for settings where elements follow a uniform probability distribution. In addition, methods exist that exploit clustering of elements in order to achieve higher…