When is the product of a set of numbers greater than the sum of them? This could well be too general a question, but I'd be interested in solutions to special cases too. Say you have some finite set of positive real numbers $x_i$, when is it the case that $\sum_i x_i > \prod_i x_i$? And when are they equal?
The special case that prompted this was an argument about whether any number is equal to the sum of its prime factors.
Any references or quick proofs welcome.
 A: The "special case" is not a special case, since only squarefree numbers equal to the product of their prime factors (I guess you forgot that primes can occur with multiplicities), and the product of a finite multiset of integers > 1 is always greater or equal to their sum, with equality only if the multiset is [2, 2] (proof by induction). So it is not really clear to me what you actually want.
A: If you have a set of positive integers (that is, no duplicates are allowed) then the sum is greater than the product if and only if the set is of the form {1,x}. The sum is equal to the product only for singleton sets {x} and the set {1,2,3}.
For, examining the remaining cases:


*

*If the set is empty the sum is 0 and the product is 1, so sum < product

*If the set has two elements {x,y}, neither of which is 1, then $xy\ge 2\max(x,y)>x+y$.

*If the set has three elements {1,2,x}, with $x>3$, the sum is $x+3$ and the product is the larger number $2x$.

*If the set has any other three elements then its product is at least three times its max and its sum is less than that.

*If the set has {1,2,3,x} then the product is 6x and the sum is x+6, smaller for all $x\ge 4$.

*If the set has any other form with $k>3$ elements then by induction the sum of the smallest $k-1$ items is less than their product. Multiplying or adding the largest item doesn't change the inequality.

