Question

a) Show that 6, 28, 496, 8128, and 33550336 are perfect numbers (recall, according to the note: n is said to be perfect if σ(n) = 2n).

b) Recall that prime numbers of the form M_n := 2ⁿ − 1 are called the Mersenne primes. For those nsuch that M_n := 2ⁿ − 1 is prime,

prove that the number P_n := 1/2 (M_n + 1)M_n= 2^(n-1)(2ⁿ − 1) is a perfect number (Note: for P₁ = 6, P₂ = 28, P₃ = 496, P₄ = 8128, P₅ = 33550336 which recover the perfect numbers in (a)).

c) Let P = q · 2^(n-1) where q is an odd prime. Prove that if P is a perfect number, then q = 2ⁿ − 1, i.e. all perfect number of the form P = q · 2^(n-1) is of the form 2^(n-1) (2ⁿ − 1).

Answer 1

Printing Mistake in the questions of (b),(c). I know the definition of Mersenne Prime .Correcting it I give the solutions.