Dr. Kenneth A. Ribet
Professor Emeritus
Department of Mathematics
UC Berkeley

The perfect squares (1, 4, 25, 36 and so forth) were studied extensively by Leonardo Fibonacci in his thirteenth century publication "The Book of Squares." Fibonacci studied triples of squares such as 1, 25, 49 for which the middle number is the average of the other two. Notice that 24, the difference between 1 and 25, is the same as the difference between 25 and 49. Fibonacci used the word "congruum" for a number that occurs as such a difference. Deciding whether or not a given number is a congruum remains an open problem. This problem is illuminated by conjectures in number theory that have been central for the last 60 years. I will describe the conjectures and explain their relation with Fibonacci's work.