Volume 11, Issue 6

We show that a completely multiplicative automatic function which does not have 0 as a value is almost periodic.

It is well known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational α . In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers α . These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r -Fibonacci sequences, and the Pell sequence, to name a few. We show that these sequences satisfy a linear recurrence relation when considered modulo k , even though the sequences themselves do not. We then employ this recurrence relation to determine the generating functions and Binet-like formulas. We end by discussing the convergence of the ratios of the terms of these sequences.

For n > 1, let , where σ ( n ) is the sum of the divisors of n . We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G ( N ) ≥ max( G ( N / p ), G ( aN )), for all prime factors p of N and each positive integer a . The proof uses Robin's and Gronwall's theorems on G ( n ). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erdős's results.

An open conjecture of Erdős and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1 + 2 = 3. Reducing the equation modulo k and k 2 , we give necessary and sufficient conditions on solutions to the resulting congruence and supercongruence. A corollary is a new proof of Moser's result that the conjecture is true for odd exponents n . The proofs use divisibility properties of power sums as well as Lerch's relation between Fermat and Wilson quotients. Examples are provided using primary pseudoperfect numbers.

The Stern sequence (also known as Stern's diatomic sequence ) is defined by s (0) = 0, s (1) = 1, and for all n ⩾ 1 by s (2 n ) = s ( n ), s (2 n + 1) = s ( n ) + s ( n + 1). In a recent paper, Roland Bacher introduced the twisted Stern sequence given by the recurrences t (0) = 0, t (1) = 1, and for n ⩾ 1 by t (2 n ) = – t ( n ), t (2 n + 1) = – t ( n ) – t ( n + 1). Bacher conjectured three identities concerning Stern's sequence and its twist. In this paper, we prove Bacher's conjectures.

Let G be an abelian group of order n of the form G ≅ ℤ n 1 ⊕ ℤ n 2 ⊕ ⋯ ⊕ ℤ n r , where n i | n i +1 for 1 ≤ i < r and n 1 > 1. Let A = {1, –1}. Given a sequence S with elements in the given group G and of length n + k such that the natural number k satisfies , where r′ = |{ i ∈ {1, 2, …, r } : 2 | n i }|, if S does not have an A -weighted zero-sum subsequence of length n , we obtain a lower bound on the number of A -weighted n -sums of the sequence S . This is a weighted version of a result of Bollobás and Leader. As a corollary, one obtains a result of Adhikari, Chen, Friedlander, Konyagin and Pappalardi. A result of Yuan and Zeng on the existence of zero-smooth subsequences and the DeVos–Goddyn–Mohar Theorem are some of the main ingredients of our proof.

We establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo p k ( k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the p -adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p -th Fourier coefficient of a particular modular form.

By means of Liouville's theorem, we show that Euler's pentagonal number theorem implies the Jacobi triple product identity.

We prove that if a subset of a d -dimensional vector space over a finite field with q elements has more than q d –1 elements, then it determines all the possible directions. We obtain a complete characterization if the size of the set is ≥ q d – 1 . If a set has more than q k elements, it determines a k -dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a k -dimensional hyperplane shows. We can view this question as an Erdős type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. For discrete subsets of , this question has been previously studied by Pach, Pinchasi and Sharir.

In 2002, F. Luca and P. G. Walsh studied the Diophantine equations of the form ( a k – 1)( b k – 1) = x 2 , for all ( a, b ) in the range 2 ≤ b < a ≤ 100 with sixty-nine exceptions. In this paper, we solve two of the exceptions. In fact, we consider the equations of the form ( a k – 1)( b k – 1) = x 2 , with ( a, b ) = (13, 4), (28, 13).

We generalize the original tennis ball problem to the case in which the number of balls received and the number of balls discarded can vary from move to move, and derive formulas solving the problem in that case. We then use these formulas to obtain new and simpler proofs of the relation between the solution to the original tennis ball problem and the Catalan numbers, as well as a considerably simpler proof for the generalization in which a fixed number s of balls are received in each move, and one ball is discarded.

We answer a question on the conditioned binomial coefficients raised in an article of Barlotti and Pannone, thus giving an alternative proof of an extension of Frobenius' generalization of Sylow's theorem.

We prove a new convolution identity for sums of products of two Bernoulli polynomials. This can be rewritten to obtain a reciprocity relation for a related sum. The proof uses some results on Stirling numbers of both kinds which are of independent interest. In particular, a class of polynomials related to the Stirling numbers of the second kind turns out to be a useful tool.

We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and asymptotics for large alphabet size and large total word length are discussed. Moreover, we derive a Gaussian limit law for the number of words in a random finite language.