Rational Amicable Pair

A rational amicable pair consists of two positive integers and for which the divisor functions are equal and are of the form

where and are bivariate polynomials with , and for which the following properties hold (Y. Kohmoto):

1. The numerator is a homogeneous polynomial.

2. The denominator is a homogeneous polynomial.

3. The polynomial degree of is one greater than that of .

If , let and . Since and are homogeneous polynomials, (1) reduces to

If , then is a perfect number. If is an integer greater than 2, then is a multiperfect number.

Consider rational functions of the form

where is a positive integer. For , (3) reduces to

whose only positive integer solution is . For , (3) reduces to

so distinct and form an amicable pair. For , (3) becomes

Kohmoto has found the following families and individual solution of this type. The first family is

where is a Mersenne prime and , giving (26403469440047700, 30193441130006700), (7664549986025275200, 8764724625167659200), ... (OEIS A038362 and A038363). The second set of solutions is

where . For , this gives the solution

A further solution is

Considering rational functions of the more general form

where and is a positive integer, Kohmoto has found the solution

for the index of a Mersenne prime with the exceptions of and 3.

Y. Kohmoto (pers. comm., Feb. 2004) also found the solution

for the index of a Mersenne prime with the exception of .

Considering rational functions of the form

for , Kohmoto has found the solution

Considering rational functions of the form

where , or equivalently,

Kohmoto has found the solutions listed in the following table.