Claire Levaillant, USC

Abstract : An Egyptian fraction is a decomposition of a rational into a sum of distinct unit fractions. These fractions appear in some of the oldest mathematical manuscripts in Egypt in 1650 BC and regained interest with Erdös in the mid 50's. Fibonacci in 1202 was the first to provide a greedy algorithm in order to obtain such decomposition for any choice of rational number.
While several other general algorithms grew over the years to attempt to minimize the length and largest integer of the decomposition, no general algorithms were ever provided to generate all the decompositions of a given length.
 In this talk, we focus on the decomposition of the unit and offer a general algorithm to find all the decompositions of a given length involving only primes 2 and q as factors in the denominators, with q any set odd prime and imposing that the exponents of 2 are less than or equal to 2. We also provide a general algorithm for counting these solutions for a given length of the decomposition.