Suppose that we have a integer m, and we need to choose n ≤ m...

woowheedr

woowheedr

Answered

2022-07-01

Suppose that we have a integer m, and we need to choose n m and x 1 , x 2 , . . . , x n such that i = 1 n x i = m and i = 1 n x i is maximized, where n and all x i 's are integers. In particular, is the maximum product i x i polynomial in m?

Answer & Explanation

Karissa Macdonald

Karissa Macdonald

Expert

2022-07-02Added 12 answers

Nope, since it is not even bounded by a polynomial : if m is even, for example, you may take the decomposition x i = 2 and n = m / 2 to obtain a product 2 m / 2 wich exceeds polynomial growth (and I'm not claiming this is the maximum, but it is a lower bound).
Crystal Wheeler

Crystal Wheeler

Expert

2022-07-03Added 4 answers

I think you'll find the product is maximized by using as many threes as possible, topping up with a two or two.

EDIT: An explanation, as requested.

First show that if there's an i such that x i 5 then you can increase the product by replacing x i by two numbers that add up to x i . Also, replacing 4 by two 2s doesn't change sum or product, so we may assume all x i are 2 or 3. Now note that if you have three 2s you can do better with two 3s.

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