Results 1 to 2 of 2

Math Help - (3^m)(5^n)(11^k) is not a perfect number

  1. #1
    Newbie
    Joined
    Apr 2012
    From
    NJ
    Posts
    5

    (3^m)(5^n)(11^k) is not a perfect number

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8

    Re: (3^m)(5^n)(11^k) is not a perfect number

     \sigma(3^m5^n11^k) = \sum_{i=0}^m \sum_{j=0}^n \sum_{l=-}^k 3^i5^j11^k
    use geometric series to evaluate this sum.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Perfect number
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: May 30th 2011, 07:58 AM
  2. Perfect Number!!! Help
    Posted in the Discrete Math Forum
    Replies: 10
    Last Post: November 16th 2009, 10:29 AM
  3. Odd perfect number
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: June 13th 2009, 07:58 PM
  4. Perfect Number
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: February 15th 2009, 07:08 PM
  5. Divisor of perfect number is not perfect
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 18th 2008, 01:26 PM

Search Tags


/mathhelpforum @mathhelpforum