When is the product of a set of numbers greater than the sum of them? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T07:54:30Z http://mathoverflow.net/feeds/question/16684 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them When is the product of a set of numbers greater than the sum of them? Seamus 2010-02-28T15:10:14Z 2012-03-06T16:18:22Z <p>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?</p> <p>The special case that prompted this was an argument about whether any number is equal to the sum of its prime factors.</p> <p>Any references or quick proofs welcome.</p> http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them/16687#16687 Answer by David Eppstein for When is the product of a set of numbers greater than the sum of them? David Eppstein 2010-02-28T15:51:59Z 2010-02-28T18:51:41Z <p>If you have a <em>set</em> 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}.</p> <p>For, examining the remaining cases:</p> <ul> <li>If the set is empty the sum is 0 and the product is 1, so sum &lt; product</li> <li>If the set has two elements {x,y}, neither of which is 1, then $xy\ge 2\max(x,y)>x+y$.</li> <li>If the set has three elements {1,2,x}, with $x>3$, the sum is $x+3$ and the product is the larger number $2x$.</li> <li>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.</li> <li>If the set has {1,2,3,x} then the product is 6x and the sum is x+6, smaller for all $x\ge 4$.</li> <li>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.</li> </ul> http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them/16688#16688 Answer by darij grinberg for When is the product of a set of numbers greater than the sum of them? darij grinberg 2010-02-28T15:54:48Z 2010-02-28T15:54:48Z <p>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.</p> http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them/90377#90377 Answer by AGD for When is the product of a set of numbers greater than the sum of them? AGD 2012-03-06T16:18:22Z 2012-03-06T16:18:22Z <p>Hi,</p> <p>Was just looking online to see if I could find a result I did for my college course many years ago.</p> <p>i found that the product of a collection of poitive reals P is related to the sum of the same collection S via this inequality.</p> <p>P >= S multiplied by e to the power of 1 divided by e.</p> <p>Can't remember how I proved it, but it did include one of Abel's theorems.</p> <p>Happy times!</p>