In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic selection and elimination of prime number indexes on the real number line.
We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic progressions of primes $p$ such that $p+h$ is also prime for each element of the progression. In case of $h=2$ this means that under some plausible unproved hypotheses about regular distribution of the primes and other sequences in arithmetic progressions we really have arbitrarily long arithmetic progressions in the sequence of twin primes.
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape $x^3 + 2y^3$ where $y$ is small.
We suggest other models of sieve generated sequences like the Sieve of Eratosthenes to explain randomness properties of the prime numbers, like the twin prime conjecture, the lim sup conjecture, the Riemann conjecture, and the prime number theorem.
We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.
Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.
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 estimates for its curvature are provided.
We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.
In this note we describe a method for finding prime numbers as fixed points of particularly simple sequences. Some basic calculations show that success rates for identifying primes this way are over 99.9%. In particular, it seems that the set of odd primes can be obtained as fixed points of the sequence which we call A(1), the sequence of smallest divisors of triangular numbers, where the divisors are positive numbers that have not yet appeared in the sequence.
In this paper, we define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that these sequences always terminate in a power of two but can be extended infinitely to the left.
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from the integer numbers to produce formulas that generate all prime numbers in some interval.
We consider a generalization of Euclid's proof of the infinitude of primes and show that it leads to variants of the Euclid-Mullin sequence that provably contain every prime number.
We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{α,β}$, where $α>1$ is irrational and of finite type.
We consider a particular case of an analog for elliptic curves to the Mersenne problem : finding explicitely all prime power terms in an elliptic divisibility sequence when descent via isogeny is possible. We explain how this question can be related to classical problems in diophantine geometry and we compute an explicit upper bound on the index of prime power terms in magnified elliptic divisibility sequences.
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
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 discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.
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.
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.
In the present paper we construct normal numbers in base $q$ by concatenating $q$-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author.
Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is an infinity of such prime sequences. Sequences having the least possible growth rate are of particular interest. This work's focus is on prime sequences with $a=1$, $b \in \{0,1\}$, that have the smallest possible constant $c$ given $d>1$, and sequences with the smallest possible $d$, given $c=2$. In particular, we prove the existence of the four infinite ex…
In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloorαp^θ\rfloor$ with $α>0$ and $θ>1$.
We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.
We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor nα\rfloor\}$, where $α$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.
Let $P(m)$ denote the greatest prime factor of $m$. For integer $a>1$, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, $$P(a^n-1)\gg_{ε, a} n^{2-ε}$$ for any $ε>0$. We study analogues results for the corresponding divisibility sequence over the function field $\mathbb{F}_q(t)$ and for some divisibility sequences associated to elliptic curves over the rational field $\mathbb{Q}$.
Let $α,β$ be real numbers such that $α>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloorαm+β\right\rfloor\big)_{m=1}^\infty$.
The main result of the paper is that assuming that the level $θ$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(θ)$ such that there are arbitrarily long arithmetic progressions with the property that $p'=p+d$ is the next prime for each element of the progression. If $θ>0.971$, then the above holds for some $d\leq 16$.
For integer a let us consider the sequence X_a={x_0,x_1,x_2,...} defined by x_0=a, x_1=1 and, for n>=1, x_{n+1}=x_n+x_{n-1}. We say that a prime p divides X_a if p divides at least one term of the sequence. It is easy to see that every prime p divides X_1, the sequence of Fibonacci numbers. Lagarias, using a technique involving the computation of degrees of various Kummerian extensions first employed by Hasse, showed in 1985 that X_2, the set of primes dividing some Lucas number has natural density 2/3 and posed as a challenge finding the density of prime divisors of X_3. In this paper we reso…
In this article we consider the completely multiplicative sequences $(a_n)_{n \in \mathbf{N}}$ defined on a field $\mathbf{K}$ and satisfying $$\sum_{p| p \leq n, a_p \neq 1, p \in \mathbf{P}}\frac{1}{p}<\infty,$$ where $\mathbf{P}$ is the set of prime numbers. We prove that if such sequences are automatic then they cannot have infinitely many prime numbers $p$ such that $a_{p}\neq 1$. Using this fact, we prove that if a completely multiplicative sequence $(a_n)_{n \in \mathbf{N}}$, vanishing or not, can be written in the form $a_n=b_nχ_n$ such that $(b_n)_{n \in \mathbf{N}}$ is a non ultimate…
We consider a numeration system in the ring of integers ${\mathcal O}_K$ of a number field, which we assume to be principal. We prove that the property of being a prime in ${\mathcal O}_K$ is decorrelated from two fundamental examples of automatic sequences relative to the chosen numeration system: the Thue-Morse and the Rudin-Shapiro sequences. This is an analogue, in ${\mathcal O}_K$, of results of Mauduit-Rivat which were concerned with the case $K={\mathbb Q}$.
Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n<x\} \sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely, $\operatorname{lcm} \{f(p)\mid p<x\}$ where $p$ denotes a prime and obtain non-trivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of $f(p).$
Suppose that $α_1, α_2,β_1, β_2 \in\mathbb{R}$. Let $α_1, α_2 > 1$ be irrational and of finite type such that $1, α_1^{-1}, α_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number in the range $1 < c < 12/11$. In this paper, it is proved that there exist infinitely many primes in the intersection of Beatty sequences $\mathcal{B}_{α_1,β_1} = \lfloorα_1 n + β_1\rfloor, \mathcal{B}_{α_2, β_2} = \lfloorα_2 n + β_2\rfloor$ and the Piatetski-Shapiro sequence $\mathscr{N}^{(c)} = \lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the inter…
For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor αn+β\rfloor : n=1,2,3, \dots \rbrace$, where $α, β\in \mathbb{R}$ with $α>1$ is irrational and we prove an asymptotic formula for the number of primes $p$ such that $g(p)=\lfloor αn+β\rfloor.$ Next we obtain an asymptotic formula for number of primes $p$ of the form $p=\lfloor αn+β\rfloor$ which also satisfies $p \equiv f \pmod d$ where $f, d$ are integers with $1\leq…
The Somos-5 sequence is defined by $a_{0} = a_{1} = a_{2} = a_{3} = a_{4} = 1$ and $a_{m} = \frac{a_{m-1} a_{m-4} + a_{m-2} a_{m-3}}{a_{m-5}}$ for $m \geq 5$. We relate the arithmetic of the Somos-5 sequence to the elliptic curve $E : y^{2} + xy = x^{3} + x^{2} - 2x$ and use properties of Galois representations attached to $E$ to prove the density of primes $p$ dividing some term in the Somos-5 sequence is equal to $\frac{5087}{10752}$.
We present a constant and a recursive relation to define a sequence $f_n$ such that the floor of $f_n$ is the $n$th prime. Therefore, this constant generates the complete sequence of primes. We also show this constant is irrational and consider other sequences that can be generated using the same method.
Let $k \ge 2$ and $α_1, β_1, ..., α_k, β_k$ be reals such that the $α_i$'s are irrational and greater than 1. Suppose further that some ratio $α_i/α_j$ is irrational. We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \mathcal B_i = \left\{n \in \mathbb N : n = [ α_i m + β_i ] \text{for some} m \in \mathbb Z \right\}. $$
Given positive integers a,b,c and d such that c and d are coprime we show that the primes p=c(mod d)dividing a^k+b^k for some k>=1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
Fix irrational numbers $α,\hatα>1$ of finite type and real numbers $β,\hatβ\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hatαm+\hatβ\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecutive primes for which $p\in B$ and $p^\sharp\in\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \big|\{p\le x:p\in B\text{ and }p^\sharp\in\hat B\}\big|=(α\hatα)^{-1}π(x)+O\big(x(\log x)^{-3/2+ε}\big), $$ where $π(x)$ is the prime cou…
For every integer $n\ge 1$ let $a_n$ be the smallest positive integer such that $n+a_n$ is prime. We investigate the behavior of the sequence $(a_n)_{n\ge 1}$, and prove asymptotic results for the sums $\sum_{n\le x} a_n$, $\sum_{n\le x} 1/a_n$ and $\sum_{n\le x} \log a_n$.
Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine equation $u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$ with some conditions on $n_i $ for all $1\leq i \leq t$. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due t…
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 terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.
