Number Theory Perfect Number Proof

[mathmajor23]

Homework Statement

Show that a number of the form 3^m5ⁿ11^k can never be a perfect number.

Any ideas?

 

[jedishrfu]

whats the definition of a perfect number?

and in this case what would be the expression for it then see if you can make one side equal to the othe
as an exampler:

6 = 1 * 2 * 3 and 6=1 + 2 + 3 therefore 6 is a perfect number.

 

[Joffan]

What does consideration of the various prime powers considered mod 4 tell you?

What does that say about the ability of the various factor sums to contain suitable powers of 3?


{jedishfru, I hope we're well beyond definitions, but the key is that the sum of all factors is twice the number considered, when one includes the number itself as a factor, which is far more convenient for generating the factor sum - see this link for a quick intro}

 

[mathmajor23]

whats the definition of a perfect number?

and in this case what would be the expression for it then see if you can make one side equal to the othe
as an exampler:

6 = 1 * 2 * 3 and 6=1 + 2 + 3 therefore 6 is a perfect number.

Define the function σ(n) as the sum over all divisors of n including n. A number n is perfect, if σ(n) = 2n.

 

[Joffan]

OK... so we know the form of n, and we know that σ(n)/n = 2... so I'm looking at σ(3^m), σ(5ⁿ) and σ(11^k), taken mod 4 first.

 

[jedishrfu]

What does consideration of the various prime powers considered mod 4 tell you?

What does that say about the ability of the various factor sums to contain suitable powers of 3?


{jedishfru, I hope we're well beyond definitions, but the key is that the sum of all factors is twice the number considered, when one includes the number itself as a factor, which is far more convenient for generating the factor sum - see this link for a quick intro}

My question on the definition of a perfect number was for the OP as this looks like a homework assignment and we are supposed to assist the OP in finding the answer. In the absence of any work shown, I started with a question.

I think the next question would be to ask whether the given number is even or odd and see what comes from that.

 

[Joffan]

My question on the definition of a perfect number was for the OP as this looks like a homework assignment and we are supposed to assist the OP in finding the answer. In the absence of any work shown, I started with a question.

And I started with two questions for the OP, which seemed to me at an appropriate level for the difficulty of the question.

I think the next question would be to ask whether the given number is even or odd and see what comes from that.

Up to you, buddy, although the form of the problem makes that a redundant question, I'd say. But I've had nothing back from this poster on my opening hints for avenues to progress. It was interesting to work out the answer, but I'm not going to go any further here unless and until I see some engagement from mathmajor23.