Aviezri Fraenkel's Work in Number Theory

Abstract

As well as spanning a number of decades, Aviezri Fraenkel’s mathematical work has spanned a number of areas. He is well known as an expert on combinatorial games and his contributions in that area are described by Richard Guy in the accompanying article. He has also made advances in number theory, combinatorics, and computer science. His doctoral thesis, written under the guidance of Ernst G. Straus at UCLA and awarded in 1961, concerned a theorem of Ridout in the theory of Diophantine equations [4]. At this time he was also working in the fledgling eld of computer science. His rst published paper [3] was on a method for using Mersenne primes for the design of a high speed multiplier. This was followed by one on \a very high-speed digital number sieve" [2] which was a special-purpose sieving device which worked at the rate of 10 10 numbers per minute, 1000 times faster than the then state of the art IBM7090. This was before people talked about computational complexity but his interest in the area remained, particularly applied to his work on games. Other early work dealt with questions about transcendental numbers, Diophantine approximation, and Diophantine equations. Another early paper was with Joe Gillis [10] on the avoidability of repetitions in the DNA code. This foreshadowed his later work on molecular biology and sequences. A few years later he published \The bracket function and complementary sets of integers"[6]. The bracket function is the integer part or floor function, at that time written [x], now usually bxc : Two sets of integers are complementary if they are disjoint and their union is the set of positive integers Z + . An old result is that the sets fbnc : n 2 Z + g and fbnc : n 2 Z + g are complementary if and only if and are irrational and 1 + 1 =1 :