Let P (m) denote the greatest prime factor of m. For integer a > 1, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, P (a − 1) ,a n2− for any > 0. We study analogues results for the corresponding divisibility sequence over the function field Fq(t) and for some divisibility sequences associated to elliptic curves over the rational field Q. In honor of M. Ram Murt...

For n a positive integer, we show that the number of of 2n-tuples of integers that are the row and column sums of some n × n matrix with entries in {0, 1} is evenly divisible by n+1. This confirms a conjecture of Benton, Snow, and Wallach. We also consider a q-analogue for m×n matrices. We give an efficient recursion formula for this analogue. We prove a divisibility result in this context that...

The Erdős–Moser equation is a Diophantine equation proposed more than 60 years ago which remains unresolved to this day. In this paper, we consider the problem in terms of divisibility of power sums and in terms of certain Egyptian fraction equations. As a consequence, we show that solutions must satisfy strong divisibility properties and a restrictive Egyptian fraction equation. Our studies le...

Corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions. These arise because of the falsity of a much used asymptotic equivalence lemma, and they impinge on the convolution equivalence closure theorem for general infinitely divisible laws.

We consider the problem of fairly allocating a bundle of infinitely divisible goods among a group of agents with “classical” preferences. We propose to measure an agent’s “sacrifice” at an allocation by the size of the set of feasible bundles that the agent prefers to her consumption. As a solution, we select the allocations at which sacrifices are equal across agents and this common sacrifice ...

The infinite divisibility of probability distributions on the space P(R) of probability distributions on R is defined and related fundamental results such as the Lévy-Khintchin formula, representation of Itô type of infinitely divisible RPD, stable RPD and Lévy processes on P(R) are obtained. As an application we investigate limiting behaviors of a simple model of a particle motion in a random ...

The division problem consists of allocating an amount of a perfectly divisible good among a group of n agents. Sprumont (1991) showed that if agents have single-peaked preferences over their shares, then the uniform allocation rule is the unique strategy-proof, efficient, and anonymous rule. We identify the maximal set of preferences, containing the set of single-peaked preferences, under which...

This paper provides tools for the study of the Dirichlet random walk in R . We compute explicitly, for a number of cases, the distribution of the random variableW using a form of Stieltjes transform ofW instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2...

In this paper two kinds of cumulant processes are studied in a general setting. These processes generalize the cumulant of an infinitely divisible random variable and they appear as the exponential compensator of a semimartingale. In a financial context cumulant processes lead to a generalized Esscher transform. We also provide some new criteria for uniform integrability of exponential martinga...