a) Show that 6, 28, 496, 8128, and 33550336 are perfect numbers(recall, according to the note: n is said to be perfect if ?(n) =2n).
b) Recall that prime numbers of the form Mn :=2n ? 1 are called the Mersenne primes. For those nsuchthat Mn := 2n ? 1 is prime,
prove that the number Pn := 1/2 (Mn +1)Mn= 2(n-1)(2n ? 1) is a perfectnumber (Note: for P1 = 6, P2 = 28,P3 = 496, P4 = 8128, P5 = 33550336which recover the perfect numbers in (a)).
c) Let P = q · 2(n-1) where q is an oddprime. 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).
