Amicable Numbers

For a positive integer n, let s(n) denote the sum of the proper divisors of n. Two distinct numbers a,b are said to form an amicable pair if s(a)=b and s(b)=a. This somewhat arbitrary concept dates back to Pythagoras and has been studied since by both numerologists and number theorists. While over 12 million amicable pairs are now know, it is still not known if there are infinitely many. Paul Erdos was the first mathematician to prove that numbers involved in an amicable pair has asymptotic density 0. Carl Pomerance proved over three decades ago that their reciprocal sum is finite, and has recently improved understanding on the underlying result on their distribution. In a talk on April 8th open to Bowdoin faculty and students, Pomerance relates some of the colorful history of amicable numbers and some of the techniques that go into proving theorems about them.

Carl Pomerance received his B.A. from Brown University in 1966 and his Ph.D. from Harvard University in 1972 under the direction of John Tate. Currently he is the John G. Kemeny Parents Professor of Mathematics at Dartmouth College, after previous positions at the University of Georgia and Bell Labs. A number theorist, Pomerance specializes in analytic, combinatorial, and computational number theory, with applications in the field of cryptology. He considers the late Paul Erdos as his greatest influence.

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