## William Cocke and Dr. Rodney Forcade, Mathematics Department

Latin squares represent one of the oldest elements of modern algebra, despite being over- shadowed by the structurally superior subclass of nite groups. Every nite group generates a Latin square, an n by n grid wherein n distinct element appear exactly once in each row and column. This grid is a Latin square

Latin squares were studied by Euler and possess many interesting combinatorial properties. Modern research on Latin squares focuses on the completion question: Can a given partial grid, be completed to form a Latin square, assuming there are no duplicate elements in the partial structure. For example, the following grid can be completed as seen below.

A speci c class of partial Latin squares arises in number theory. Essentially, one starts with a Latin square designed as a multiplication table, with the assumption that any entry that exceeds the size of the square is left blank. Thus for n = 4 we have the following table.

This is the same table as above, and we can choose appropriate symbols and complete the Latin square. It was conjectured in 1986 that such partial Latin squares can always be completed so that the system forms the multiplication table of a group. This conjecture is called the FLP conjecture. It was soon discovered that the FLP conjecture for all n is false with the rst counterexample occurring for n = 195.

We investigated two aspects of the problem. First, we ran a serious of computational tests on the Mary{Lou super computer designed to determine what types of groups can have an underlying Latin square with the desired structure. Second, we investigated the relationship between the FLP conjecture and a weaker conjecture regarding the existence of primes of a given form with speci c properties. The computations we ran focused rst on attempting to determine just how close to group-able, complete-able as a group, the Latin square corre- sponding to 195 is and what types of results that determines. We found that if one simply erases one of the rows and columns to create a 194 by 194 table, the result is a group-able. That is, the table for 195 is in some sense as close to group-able as possible, without being group-able.

We also ran calculations to determine if the group structure from such a table is always Abelian. We found using centralizers and centers, that up to approximately n = 120; 000 the group is also always. For n between 120; 000 and 200; 000 the group structure is either Abelian or meta-Abelian.

Currently we are investigating the connection between the FLP conjecture for n = p a prime number, and the existence of primes q such that

q = nk + 1:

The kth powers of 1; : : : ; n are distinct modulo q.

The existence of such a prime implies the FLP conjecture for n = p, meaning the resulting table is group-able. However, we believe the converse direction is also true. This new project has been developed in the last two months as an alternative approach to the problem and should o er promising new insights. We have interesting heuristics, but are still working on demystifying these new results.