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}.

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 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.

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.

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.

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.

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system of linear forms on 𝔽np with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of 𝔽np. While in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153], we use the Hahn–Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the U3 inverse theorem [B. J. Green and T. Tao, An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153].

Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n by

If we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

Let P=A×A⊂𝔽p×𝔽p, p a prime. Assume that P=A×A has n elements, n<p. See P as a set of points in the plane over 𝔽p. We show that the pairs of points in P determine lines, where c is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of n points and a set of n lines in the projective plane over 𝔽p (n<p)is bounded by , where C is an absolute constant.

Here, we show that for most primes p, every residue class modulo p can be represented as a sum of 32 Fibonacci numbers.

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [2, 1000].

Erdős and Szekeres [‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz.5 (1978), 97–99] showed that for any four positive integers satisfying m1+m2=n1+n2, the two binomial coefficients (m1+m2)!/m1!m2! and (n1+n2)!/n1!n2! have a common divisor greater than 1. The analogous statement for k-element families of k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman.

Erdős and Szekeres remark that if m1,m2,n1,n2 as above are all greater than 1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m1 +m2 .Such a bound is obtained in Section 2.

The remainder of this paper is devoted to proving results that narrow the class of possible counterexamples to Wasserman’s conjecture.