A subset D of a finite cyclic group ℤ/mℤ is called a perfect difference set if every nonzero element of ℤ/mℤ can be written uniquely as the difference of two elements of D. If such a set exists, then a simple counting argument shows that m=n2+n+1 for some non-negative integer n. Singer constructed examples of perfect difference sets in ℤ/(n2+n+1)ℤ whenever n is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists. In this talk, I will discuss a proof of an asymptotic version of this conjecture: the number of n<N for which ℤ/(n2+n+1)ℤ contains a perfect difference set is ~N/log(N).
This talk is related to this arXiv paper.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
