Define P(n) as the sum of all the positive integers that are less than n and relatively prime to n.
Find all possible positive integer values of n such that P(n) = 3n, and prove that there are no others.
I'm not sure this works but I think it does:
For any n, if k < n is relatively prime to n, then so is n-k.
Therefore for any n, you can create pairs of numbers (k, n-k) which are both relatively prime to n and obviously sum to n.
Then the only way P(n) = 3n is where phi(n) is 6. This is the case for n = 7, 9, 14, and 18.
The last step would be proving that phi(n) != 6 for all other integers.
Posted by tomarken
on 2014-06-12 12:29:35