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 Product equals sum (Posted on 2016-01-17)
Let's start with an easy problem : What three positive integers have a sum equal to their product?

This puzzle can easily be transformed into a D4 problem:

For what values of k will the question "What k positive integers have a sum equal to their product?" have only one unique set of integers for an answer?
Clearly for k=2 the answer is unique: (2,2) and so it is for k=4: (1,1,2,4).

List all other values of k below 1000.

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 A start | Comment 2 of 6 |
For starters, let's consider where only 2 of the numbers are not 1.  List the values in descending sequence

First up, (m,2,1,...,1).  The product = 2m, so the sum equals the product if there are 2m - m - 2 ones.
In which case there are m total terms.  So one of these series works for all k = m, where m >= 2

Next up, (m,3,1,...,1).  The product = 3m, so the sum equals the product if there are 3m - m - 3 ones.
In which case there are 2m - 1 total terms.  So one of these series works for all k = 2m-1, where m >= 3.
In other words, k in (5,7,9,11 ...).  So, there are no odd k that are unique greater than 3.

Next up, (m,4,1,...,1).  The product = 4m, so the sum equals the product if there are 4m - m - 4 ones.
In which case there are 3m - 2 total terms.  So one of these series works for all k = 3m-2 where m >= 4.
In other words, k in (10,13,16,19, ...).

In fact, just by considering (m, n, 1,....1) we can rule out any k = (n-1)m - (n-2) where m >= n
When n = 5, we can rule out (17,21,25,29...)  These are all odd and have already been ruled out already
there is no need to consider odd n
When n = 6, we can rule out (26, 31, 36, 41, 46, ...)
When n = 8, we can rule out (50, 57, 64, 71, etc.)

Do far, we are down to 2,3,4,6,8,12,14,18,20,24,20,32,38,42,44,48,54 (up through 60)

I guess that we need to consider more terms than 2 that are not equal to 1

 Posted by Steve Herman on 2016-01-17 13:21:10

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