Amicable pairs and aliquot cycles for elliptic curves

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j not 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grossencharacter evaluated at a prime ideal P in End(E) having the property that #E(F_P) is prime. This is especially intricate for the family of curves with j = 0.Comment: 53 page

Some results concerning exponential divisors

If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)(n)=2n; (m;n) is said to be an e-amicable pair if σ(e)(m)=m+n=σ(e)(n); n0,n1,n2,… is said to be an e-aliquot sequence if ni+1=σ(e)(ni)−ni. Among the results established in this paper are: the density of the e-perfect numbers is .0087; each of the first 10,000,000e-aliquot sequences is bounded

Amicable pairs : a survey

In 1750, Euler [20, 21] published an extensive paper on amicable pairs, by which he added fifty-nine new amicable pairs to the three amicable pairs known thus far. In 1972, Lee and Madachy [45] published a historical survey of amicable pairs, with a list of the 1108 amicable pairs then known. In 1995, Pedersen [48] started to create and maintain an Internet site with lists of all the known amicable pairs. The current (February 2003) number of amicable pairs in these lists exceeds four million. The purpose of this paper is to update the 1972 paper of Lee and Madachy, in order to document the developments which have led to the explosion of known amicable pairs in the past thirty years. We hope that this may stimulate research in the direction of finding a proof that the number of amicable pairs is infinite