What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments

Moser asked for a construction of explicit tournaments on $n$ vertices having at least $(\frac{n}{3e})^n$ Hamilton cycles. We show that he could have asked for rather more

Almost Odd Random Sum-Free Sets

We show that if S_1 is a strongly complete sum-free set of positive integers, and if S_0 is a finite sum-free set, then with positive probability a random sum-free set U contains S_0 and is contained in S_0\cup S_1. As a corollary we show that with positive probability, 2 is the only even element of a random sum-free set

Elliptic curves, modular forms, and sums of Hurwitz class numbers

Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime, then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper, we investigate the behavior of this sum with the additional condition $r\equiv c\pmod m$. Three different methods will be explored for determining the values of such sums. First, we will count isomorphism classes of elliptic curves over finite fields. Second, we will express the sums as coefficients of modular forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke operators to obtain Hurwitz class number relations. The cases $m=2,3$ and 4 are treated in full. Partial results, as well as several conjectures, are given for $m=5$ and 7.Comment: Preprint of an old pape

FINITE FIELD ELEMENTS OF HIGH ORDER ARISING FROM MODULAR CURVES (APPEARED IN DESIGNS, CODES, AND CRYPTOGRAPHY)

Abstract. In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result. 1

A Curious Binomial Identity

1991 Mathematics Subject Classification. 05A10.In this note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients

Dependent Sets of Constant Weight Binary Vectors

We determine lower bounds for the number of random binary vectors, chosen uniformly from vectors of weight k, needed to obtain a dependent set