Ezra Brown

Doubly regular tournaments are equivalent to skew hadamard matrices

Abstract Doubly regular tournaments and homogeneous tournaments are defined and shown to be equivalent. Existence of such a tournament of order n is equivalent to the existence of a skew Hadamard matrix of order n + 1. Thus, such a tournament of order n exists whenever n + 1 = 2 t k 1 , … k s , each k i of the form p r + 1 ≡ 0 (mod 4), p a prime, and for various other orders.

Cycles of directed graphs defined by matrix multiplication (mod n )

Abstract Let A be a k × k matrix over a ring R; let GM( A , R ) be the digraph with vertex set R k , and an arc from v to w if and only if w = Av . In this paper, we determine the numbers and lengths of the cycles of GM( A , R ) for k =2 in the following two cases. (a) R= F q , the q-element finite field, and (b) R= Z /n Z and GCD( n ,det( A ))=1. This extends previous results for k =1 and R= Z /n Z . We make considerable use of the Smith form of a matrix; other than that, the most powerful tool we use is the Chinese Remainder Theorem.

The power of 2 dividing the class-number of a binary quadratic discriminant

Abstract In this paper we study congruence conditions on class numbers of binary quadratic discriminants d , modulo powers of 2, where d has two or three distinct prime divisors.

Binary quadratic forms of determinant −pq

Abstract The following theorem is proved: If p and q are distinct primes of the form 4 n + 1, and ( p | q ) = 1, then x 2 − pqy 2 represents −1 if ( p | q ) 4 = ( q | p ) 4 = −1; p if ( q | p ) 4 = −( p | q ) 4 = 1; q if ( p | q ) 4 = −( q | p ) 4 = 1. If ( p | q ) 4 = ( q | p ) 4 = 1, there are examples with any of the three being represented.

Three Fermat Trails to Elliptic Curves

Ezra (Bud) Brown (brown@math.vt.edu) professes mathematics at Virginia Tech, where he has been since 1969. The elliptic curve bug first bit him while he was in graduate school at Louisiana State, and has never really gone away. Although his main research has been in quadratic forms and algebraic number theory, he once wrote a paper with a sociologist. He loves to talk about mathematics and its history with anyone, especially students. He occasionally sings in operas, plays jazz piano just for fun, and bakes biscuits for his classes. He and his mathematical grandfather, L. E. Dickson, have the same birthday.

Why Ellipses Are Not Elliptic Curves

Summary Elliptic curves are a fascinating area of algebraic geometry with important connections to number theory, topology, and complex analysis. As their current ubiquity in mathematics suggests, elliptic curves have a long and fascinating history stretching back many centuries. This paper presents a survey of key points in their development, via elliptic integrals and functions, closing with an explanation of why no elliptically-shaped planar curved line may ever be called an elliptic curve.

Kirkman's Schoolgirls Wearing Hats and Walking through Fields of Numbers

Imagine fifteen young ladies at the Emmy Noether Boarding School—Anita, Barb, Carol, Doris, Ellen, Fran, Gail, Helen, Ivy, Julia, Kali, Lori, Mary, Noel, and Olive. Every day, they walk to school in the Official ENBS Formation, namely, in five rows of three each. One of the ENBS rules is that during the walk, a student may only talk with the other students in her row of three. These fifteen are all good friends and like to talk with each other—and they are all mathematically inclined. One day Julia says, “I wonder if it’s possible for us to walk to school in the Official Formation in such a way that we all have a chance to talk with each other at least once a week?” “But that means nobody walks with anybody else in a line more than once a week,” observes Anita. “I’ll bet we can do that,” concludes Lori. “Let’s get to work.” And what they came up with is the schedule in TABLE 1.

Why is PSL(2,7)≅ GL(3,2)?

(2009). Why is PSL(2,7)≅ GL(3,2)? The American Mathematical Monthly: Vol. 116, No. 8, pp. 727-732.

Class numbers of complex quadratic fields

Abstract Congruence conditions on the class numbers of complex quadratic fields have recently been studied by various investigators, including Barrucand and Cohn, Hasse, and the author. In this paper, we study the class number of Q (√ − pq ), where p ≡ q (mod 4) are distinct primes.

Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves

Ezra (Bud) Brown (brown@math.vt.edu) has degrees from Rice and Louisiana State, and has been at Virginia Tech since the first Nixon Administration. His research interests include graph theory, the combinatorics of finite sets, and number theory?especially elliptic curves. In 1999, he received the MAA MD-DC-VA Section Award for Outstanding Teaching, and he loves to talk about mathematics and its history to anyone, especially students.