positive whole numbers

Question

Given a list of positive whole numbers whose sum is 100 how large can their product be

Answer 1

Hi Mahir M.

That depends on whether the positive integers must be all different. If not, 3 repeated 33 times, and a 1, max you out at log(n) ~ 15.745 . Now if you must have all different integers, a little experimentation should convince you that you have a progressive penalty for factors >3, so you'd like to keep your list as "low" as possible (except, without 1, which doesn't get you anywhere!). But if you do 2 ... 13 you are cut off at sigma = 90. So drop the 4, add the 14, and you have exactly sigma = 100, log(n) ~ 10.338 . Not so large by comparison, but there you are.

Now for a related problem, which requires a bit more calculation:

What is the next larger integer than 6^100000 which possesses only 2's and 3's as factors? (Or, more to the point, how many 2's and 3's, since the actual number is not easily expressible)?

Comment back if you want to get tips on solving this latter problem!

-- Cheers, -- Mr. d.