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

(7)

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

(8)

where ,
giving the solution

(9)

The third type is the unique solution

(10)

(11)

Considering polynomials of the more general form

(12)

Kohmoto has found the solution

(13)

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

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

(14)

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