Band 11, Heft 4

We study the question of whether for each n there is an m ≠ n such that λ ( m ) = λ ( n ), where λ is Carmichael's function. We give a “near” proof of the fact that this is the case unconditionally, and a complete conditional proof under the Extended Riemann Hypothesis.

Csikvári, Gyarmati, and Sárközy asked whether, whenever the set ℕ of positive integers is finitely colored, there must exist monochromatic a, b, c , and d with a ≠ b such that a + b = cd . We provide an affirmative answer, showing that a much stronger statement is true.

The Fibonacci Zeta functions are defined by . Several aspects of the function have been studied. In this article we generalize the results by Ohtsuka and Nakamura, who treated the partial infinite sum for all positive integers n .

A class of partitions that exhibit substantial symmetry, called recursively self-conjugate partitions, are defined and analyzed. They are found to have connections to non-squashing partitions and other combinatorial objects.

Let be the multiplicative group of a finite field 𝔽 q of cardinality q , with q odd; let denote its power set. We define the arithmetical function via D ( S ) = # I ( x + x –1 , S ) – # I ( x – x –1 , S ), where for , I ( x ± x –1 , S ) = { x ± x –1 : x ∈ S }. Furthermore, let and let F ( k, l ) be the coefficient of x l in ( x –1 + 6 + x ) k . Then

We consider a finite set of lattice points and their convex hull. The author previously gave a geometric proof that the sumsets of these lattice points take over the central regions of dilated convex hulls, thus revealing an interesting connection between additive number theory and geometry. In this paper, we will see an algebraic proof of this fact when the convex hull of points is a simplex, exploring the connection between additive number theory and geometry further.

We establish a finite analogue of the Erdős–Fuchs theorem, showing that the representation function of a non-trivial subset of a finite abelian group cannot be nearly constant. Our results are, essentially, best possible.

Let C n and C n ( q ) be the n th Catalan number and the n th q -Catalan number, respectively. In this paper, we show that the Diophantine equation C n = C m ( q ) has only finitely many integer solutions ( m, n, q ) with m > 1, n > 1, q > 1. Moreover, they are all effectively computable.

Let A be an infinite set of generators for a group G , and let L A ( r ) denote the number of elements of G whose word length with respect to A is exactly r . The growth function L A is a function from the nonnegative integers ℕ 0 to the set ℕ 0 ∪ {∞}. The purpose of this note is to determine all growth functions associated to infinite generating sets for groups.

Suppose g ≥ 2. A natural number N is called a repdigit in base g if it has the shape for some 1 ≤ a < g , i.e., if all of its digits in its base g expansion are equal. The number N is called perfect if σ ( N ) = 2 N , where σ ( N ) ≔ Σ d | N d is the usual sum of divisors function. We show that in each base g , there are at most finitely many repdigit perfect numbers, and the set of all such numbers is effectively computable. In particular, 6 is the only repdigit perfect number in base 10.

We establish a numerically explicit version of the Pólya–Vinogradov inequality for the sum of values of a Dirichlet character on an interval. While the technique of proof is essentially that of Landau from 1918, the result we obtain has better constants than in other numerically explicit versions that have been found more recently.

The height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic polynomials. One result, due to H. Maier, gives a best possible upper bound of n ψ ( n ) for almost all n , where ψ ( n ) is any function that approaches infinity as n → ∞. We will discuss the related problem of bounding the maximal height over all polynomial divisors of x n – 1 and give an analogue of Maier's result in this scenario.

We prove a pair of identities expressing Bernoulli numbers and Bernoulli numbers of the second kind as sums of generalized falling factorials. These are derived from an expression for the Mahler coefficients of degenerate Bernoulli numbers. As corollaries several unusual identities and congruences are derived.

In this paper we provide an algebraic derivation of the explicit Witten volume formulas for a few semi-simple Lie algebras by combining a combinatorial method with the ideas used by Gunnells and Sczech in the computation of higher-dimensional Dedekind sums.