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Joint Berkeley-Caltech-Stanford Number Theory Seminar

Monday, November 2, 2020
12:30pm to 2:00pm
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An asymptotic version of the prime power conjecture for perfect difference sets
Sarah Peluse, Department of Mathematics, Princeton University/IAS,

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=n^2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in Z/(n^2+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 less than N for which Z/(n^2+n+1)Z contains a perfect difference set is ~N/log(N).

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].