1
$\begingroup$

I formulated the following claim after reading Problem 88 of Project Euler:

Fix $k$ and let $\mathscr N$ be the set of numbers $N$ satisfying $$N=n_1+n_2+\cdots+n_k=n_1n_2\cdots n_k,$$ where each $n_i$ is a positive integer.

Claim: $\mathscr N$ is finite for every $k>1$.

I believe that it is true, but I do not know how to prove it.

On a side note, I proved that $\mathscr N$ is nonempty for every $k>1$:

$$\begin{align}N=2+k+\overbrace{1+1+\cdots+1}^{k-2\text{ times}}=2k&\implies2+k+k-2=2k\\&\implies2k=2k\\&\implies N\in\mathscr N.\end{align}$$

Could it be that $2k$ bounds $N$?

$\endgroup$
2
$\begingroup$

Wlog. $n_1\ge n_2\ge\ldots \ge n_k$. We cannot have $n_1=1$ as that would lead to $k\cdot 1=1^k$: Let $m$ be the largest index with $n_m>1$. We have $m=1$ would lead to $n_1+k-1=n_1$, so we must have $m\ge 2$. Then, supposing $n_1>k$, $$n_1+\ldots+n_m+1+\ldots +1\le mn_1+k-m<n_1(m+1)-1$$ $$n_1\cdots n_m\cdot 1\cdots 1\ge n_12^{m-1}$$ But then $2^{m-1}<m+1$. Conlcude $m\le 2$ and continue from there

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.