Volume 11, Issue 5

A method is given for constructing congruent numbers with three prime factors of the form 8 k + 3. A family of such numbers is given for which the Mordell–Weil rank of their associated elliptic curves equals 2, the maximal rank and expected rank for a congruent number curve of this type.

Bhargava defined p -orderings of subsets of Dedekind domains and with them studied polynomials which take integer values on those subsets. In analogy with this construction for subsets of ℤ ( p ) and p -local integer-valued polynomials in one variable, we define projective p -orderings of subsets of . With such a projective p -ordering for we construct a basis for the module of homogeneous, p -local integer-valued polynomials in two variables.

We examine the impartial combinatorial game Babylon. We abstract the game so that it is suitable to combinatorial analysis, and present a full characterization and strategies for the cases with odd number of tokens and only two colors. We also demonstrate partial results on games with even tokens with two colors, an initial extension to three colors, and offer directions for future work.

In this article we will show that for every natural d and n > 1 there exists a natural number t such that for every d -dimensional simplicial complex with vertices in , the number of lattice points in the t th dilate of is exactly modulo n , where is the Euler characteristic of .

We study the previously introduced bracketed tiling construction and obtain direct proofs of some identities for the Fibonacci and Lucas numbers. By adding a new type of tile we call a superdomino to this construction, we obtain combinatorial proofs of some formulas for the Fibonacci and Lucas polynomials, which we were unable to find in the literature. Special cases of these formulas occur in the text by Benjamin and Quinn, where the question of finding their combinatorial proofs is raised. In the process, we also show, via direct bijections, that the bracketed (2 n )-bracelets as well as the linear bracketed (2 n + 1)-tilings both number 5 n .

We study a generalization of the Frobenius problem: given k positive relatively prime integers, what is the largest integer g 0 that cannot be represented as a nonnegative integral linear combination of the given integers? More generally, what is the largest integer g s that has exactly s such representations? We construct a family of integers, based on a recent paper by Tripathi, whose generalized Frobenius numbers g 0 , g 1 , g 2 , . . . exhibit unnatural jumps; namely, g 0 , g 1 , g k , , , . . . form an arithmetic progression, and any integer larger than has at least representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.

Let P ( n ) stand for the largest prime factor of n ≥ 2 and set P (1) = 1. For each integer n ≥ 2, let δ ( n ) be the distance to the nearest P ( n )-smooth number, that is, to the nearest integer whose largest prime factor is no larger than that of n . We provide a heuristic argument showing that Σ n ≤ x 1/ δ ( n ) = (4 log 2 – 2 + o (1)) x as x → ∞. Moreover, given an arbitrary real-valued arithmetic function ƒ, we study the behavior of the more general function δ ƒ ( n ) defined by δ ƒ ( n ) = min 1≤ m ≠ n , ƒ( m )≤ƒ( n ) | n – m | for n ≥ 2, and δ ƒ (1) = 1. In particular, given any positive integers a < b , we show that Σ a ≤ n < b 1/ δ ƒ ( n ) ≥ 2( b – a )/3 and that if ƒ( n ) ≥ ƒ( a ) for all n ∈ [ a , b [, then one has Σ a < n < b δ ƒ ( n ) ≤ ( b – a ) log( b – a )/(2 log 2).

We present two infinite families of sequences that are analogous to the Stern sequence. Sequences in the first family enumerate the set of positive rational numbers, while sequences in the second family enumerate the set of positive rational numbers with either an even numerator or an even denominator.

In 1992, Ma made a conjecture related to Pell equations. In this paper, we use Störmer's theorem and related results on Pell equations to prove some particular cases of Ma's conjecture.

In this note, we prove that the projective normality of (ℙ( V )/ G , ℒ), the celebrated theorem of Erdős–Ginzburg–Ziv and normality of an affine semigroup are all equivalent, where V is a finite dimensional representation of a finite cyclic group G over ℂ and ℒ is the descent of the line bundle 𝒪(1) ⊗| G | .

In a previous paper we investigated the (exponential) bipartitional polynomials involving polynomial sequences of trinomial type. Our aim is to give properties of bipartitional polynomials related to the derivatives of polynomial sequences of trinomial type. Furthermore, we deduce identities involving Bell polynomials.

For a subset A of ℕ = {0, 1, 2, . . .}, the representation function of A is defined by r A ( n ) = |{( a, b ) ∈ A × A : a + b = n }|, for n ∈ ℕ, where | E | denotes the cardinality of a set E . Its supremum is the element s ( A ) = sup{ r A ( n ) : n ∈ ℕ} of . Interested in the question “when is s ( A ) = ∞?”, we study some properties of the function A ↦ s ( A ), determine its range, and construct some subsets A of ℕ for which s ( A ) satisfies certain prescribed conditions.