Partitions of numbers and the algebraic principle of Mersenne, Fermat and even perfect numbers

Let rho be an odd prime greater than or equal to 11. In a previous work, starting from an M-cycle in a finite field F rho, it has been established how the divisors of Mersenne, Fermat and Lehmer numbers arise. The converse question has been taken up in a succeeding work and starting with a factor of these numbers, a method has been provided to find an odd prime rho and the M-cycle in F rho contributing the factor under consideration. Continuing the study of the two previous works, a certain type of partition of a natural number is considered in the present paper. Concerning the Mersenne, Fermat and even perfect numbers, the algebraic principle is established.