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A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]

We confirm three conjectures of Guo and Zeng on Franel numbers and binomial coefficients. Furthermore, we prove more refined divisibility results for a large class of polynomial sequences.

We consider a generalized type of unique factorization of the positive integers with restrictions on the exponents and view them as a family of arithmetic convolutions and divisibility relations, similar to the convolutions defined by Narkewicz [On a class of arithmetical convolutions, Colloq. Math. 10 (1963) 81–94]. We introduce special types of multiplicativity corresponding to these convolutions, and discuss algebraic properties of the associated arithmetic convolutions and analogs of the Möbius functions. We also prove asymptotics for analogs of the totient function, totient summatory function, and divisor summatory function.

In this paper, we establish some new divisibility results involving the Franel numbers [Formula: see text] [Formula: see text] and the polynomials [Formula: see text] [Formula: see text]. For example, we show that for any positive integer [Formula: see text] we have [Formula: see text] and [Formula: see text] and for any prime [Formula: see text] we have [Formula: see text]

Let [Formula: see text] with [Formula: see text] and [Formula: see text] be an odd prime. We find a supercongruence for [Formula: see text] and related sums of powers of binomial coefficients. These results complement prior results for [Formula: see text] with [Formula: see text] obtained recently by the author.

Let [Formula: see text] be an even integer and [Formula: see text] be an odd prime. We study Franel-like sums and alternating sums, as well as lacunary sums of [Formula: see text]th powers of binomial coefficients from the point of view of arithmetic properties. This paper complements the author’s prior work on the cases with [Formula: see text] odd although, it uses a different approach. It develops new supercongruences and determines the [Formula: see text]-adic order of these sums as well as of generalized harmonic sums restricted to particular remainder classes modulo [Formula: see text].

Let [Formula: see text] be an integer and [Formula: see text] be an odd prime. We study sums and lacunary sums of [Formula: see text]th powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the [Formula: see text]-adic convergence of some subsequences and that in every step we gain at least one or three more [Formula: see text]-adic digits of the limit if [Formula: see text] or [Formula: see text], respectively. These gains are exact under some explicitly given conditions. The main tools are congruential and divisibility properties of the binomial coefficients and multiple and alternating harmonic sums.

We prove that if [Formula: see text] is a Lucas sequence satisfying some mild hypotheses, then the number of positive integers [Formula: see text] does not exceed [Formula: see text] and such that [Formula: see text] divides [Formula: see text] is less than [Formula: see text] as [Formula: see text]. This generalizes a result of Luca and Tron about the positive integers [Formula: see text] dividing the [Formula: see text]th Fibonacci number, and improve a previous upper bound due to Alba González, Luca, Pomerance and Shparlinski.

Let [Formula: see text] denote the ring of polynomials over [Formula: see text], the finite field of [Formula: see text] elements. We prove an estimate for fractional parts of polynomials over [Formula: see text] satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.

Sun introduced three kinds of numbers: Sn =∑k=0n nk2 2k k (2k+1),sn =∑k=0n nk2 2k k 1 2k − 1, and Sn+ =∑ k=0n nk2 2k k (2k+1)2. In this paper, we mainly prove that 4∑k=0n−1kS k ≡∑k=0n−1s k ≡∑k=0n−1S k+ ≡ 0(modn2)forn ≥ 1, by establishing some binomial coefficient identities, such as 4∑k=0n−1kS k = n2∑ k=0n−1 1 k + 1 2k k 6k n − 1 k2 + n − 1 k n − 1 k + 1 . This confirms several recent conjectures of Sun.

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 ( mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

We extend the Lucas–Lehmer theory for second-order divisibility sequences to a large class of fourth-order sequences, with appropriate laws of apparition and of repetition. Examples are provided by the numbers of perfect matchings, or of spanning trees, in families of graphs, and by the numbers of points on elliptic curves over finite fields. Whether there are fourth-order divisibility sequences not covered by our theory is an open question.

In 1949, Lehner showed that certain coefficients of the modular invariant j(τ) are divisible by high powers of small primes. Kolberg refined Lehner's results and proved congruences for these coefficients modulo high powers of these primes. We extend Lehner's and Kolberg's work to the elements of a canonical basis for the space of weight 0 weakly holomorphic modular forms.

The paper is devoted to the study of some properties of generalized arithmetical functions extended to the case of three variables. The convolution in this case is a convolution of the incidence algebra of a Möbius category in the sense of Leroux. This category is a two-sided analogue of the poset (it is viewed as a category) of positive integers ordered by divisibility.

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space [Formula: see text], where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.

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 ak+bkfor 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.

We investigate the function b(n) = ∑ 1, where the summing is over all i for which $(n,i) > 1, i | {\fontsize{9}{11}\selectfont\Big(\!\!\begin{array}{c}n-i-1\\[2pt]i-1\end{array}\!\!\Big)}$.

Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp= -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.

In this paper, we give some heuristics suggesting that if (un)n≥0is the Lucas sequence given by un= (an- 1)/(a - 1), where a > 1 is an integer, then ω(un) ≥ (1 + o(1)) log n log log n holds for almost all positive integers n.