Let $L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $ be the $L$-series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 0} $ be a nondegenerate binary recurrence sequence. We prove that if ${ \mathcal{M} }_{E} $ is the set of $n$ such that ${a}_{n} \not = 0$ and ${ \mathcal{N} }_{E} $ is the subset of $n\in { \mathcal{M} }_{E} $ such that $\vert {a}_{n} \vert = \vert {u}_{m} \vert $ holds with some integer $m\geq 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $.

Recently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.

For $n= 1, 2, 3, \ldots $ let ${S}_{n} $ be the sum of the first $n$ primes. We mainly show that the sequence ${a}_{n} = \sqrt[n]{{S}_{n} / n}~(n= 1, 2, 3, \ldots )$ is strictly decreasing, and moreover the sequence ${a}_{n+ 1} / {a}_{n} ~(n= 10, 11, \ldots )$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.

We prove that for each positive integer k in the range 2≤k≤10 and for each positive integer k≡79 (mod 120) there is a k-step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.

Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed $l$, the number of pairs of squareful numbers $n$, $n+l$ such that $n$is less than a given quantity.

In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

In this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.

Let K be a number field. For f∈K[x], we give an upper bound on the least positive integer T=T(f) such that no quotient of two distinct Tth powers of roots of f is a root of unity. For each ε>0 and each f∈ℚ[x] of degree d≥d(ε) we prove that . In the opposite direction, we show that the constant 2cannot be replaced by a number smaller than 1 . These estimates are useful in the study of degenerate and nondegenerate linear recurrence sequences over a number field K.

In this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Let k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k{(n+i)2+1}.

We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

We prove that the sequence {log ζ(n)}n≥2 is not holonomic, that is, does not satisfy a finite recurrence relation with polynomial coefficients. A similar result holds for L-functions. We then prove a result concerning the number of distinct prime factors of the sequence of numerators of even indexed Bernoulli numbers.

For a primitive root g modulo a prime p≥1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers agn, n=1,…,N, which is uniform over all integers a with gcd (a,p)=1.

In this paper we give an extension of a curious combinatorial identity due to B. Sury. Our proof is very simple and elementary. As an application, we obtain two congruences for Fermat quotients modulo p3. Moreover, we prove an extension of a result by H. Pan that generalizes Carlitz’s congruence.

Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A)contains a term of {ak }. The lower bound for |A| is optimal. Beyond these, we also prove that if n≥3is an integer and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A)contains a power of 2. Furthermore, cannot be improved.

For a prime power q, let 𝔽q be the finite field of q elements. We show that 𝔽*q⊆d𝒜2 for almost every subset 𝒜⊂𝔽q of cardinality ∣𝒜∣≫q1/d. Furthermore, if q=p is a prime, and 𝒜⊆𝔽p of cardinality ∣𝒜∣≫p1/2(log p)1/d, then d𝒜2 contains both large and small residues. We also obtain some results of this type for the Erdős distance problem over finite fields.

A set A⊆ℤ is called an asymptotic basis of ℤ if all but finitely many integers can be represented as a sum of two elements of A. Let A be an asymptotic basis of integers with prescribed representation function, then how dense A can be? In this paper, we prove that there exist a real number c>0 and an asymptotic basis A with prescribed representation function such that for infinitely many positive integers x.

In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.