use geometric series to evaluate this sum.
Show that a number of the form (p^m)(q^n), where p and q are Distinct Odd Primes,can never be a perfect number. Also show that a number of the form (3^m)(5^n)(11^k) can never be a perfect number.
I can show that (p^m)(q^n), where p and q are Distinct Odd Primes,can never be a perfect number
Write σ(N) = Σ d [d: d|N] the sum of the divisors of n. Recall that N is perfect when σ(N) = 2N. Also recall that σ is multiplicative. If N= (p^m)(q^n) is a prefect number then 2 = σ(N)/N = {σ(p^m)/p^m}{σ(q^n)/q^n} = (1+1/p + ...+1/p^m)(1+1/q+...+1/q^n) < (1 +1/p +...)(1 + 1/q +...) = {p/(p-1)]{(q/(q-1)}. Note that p/(p-1) = 1 + 1/(p-1) is max when p-1 is min. Since p and q are distinct odd primes then the product {p/(p-1)]{(q/(q-1)} is max when p = 3 and q = 5. So {p/(p-1)]{(q/(q-1)} <= (3/2)(5/4) = 15/8 < 2, which leads to the contradiction 2 = {σ(p^m)/p^m}{σ(q^n)/q^n} < 2.
Im not sure how to prove (3^m)(5^n)(11^k) can never be a perfect number?