Rational Amicable Pair

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

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

1. All the degrees of terms of the numerator of the right fraction are the same.

2. All the degrees of terms of the denominator of the right fraction are the same.

3. The degree of is one greater than the degree of .

If and is of the form , then (◇) reduces to the special case

so if is an integer, then is a multiperfect number.

Consider polynomials of the form

For , (◇) reduces to

of which no examples are known. For , (◇) reduces to

so form an amicable pair. For , (◇) becomes

Kohmoto has found three classes of solutions of this type. The first is

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

where , giving the solution

The third type is the unique solution

Considering polynomials of the more general form

Kohmoto has found the solution

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

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

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

Considering polynomials of the form

for , Kohmoto has found the solution

Considering polynomials of the form

or equivalently,

Kohmoto has found the solutions listed in the following table.