2
$\begingroup$

Prove that for every $n\in\mathbb{N}$ there are infinitely many ways to represent $n$ as $ n=*1^2*2^2*\dots*k^2, $ where $k\in\mathbb{Z}$ and $*\in\{+,-\}$.

On my paper, this problem is marked as "Erdos-Suranyi", but I could find anything about it on Google.

  • 0
    I suspect that notation simply means that the problem (or the result) can be found in *Topics in the Theory of Numbers*, by Erdős and Surányi.2012-03-12

1 Answers 1

4

The classic solution to show existence of one solution is to use the identity

$ (n+3)^2 - (n+2)^2 - (n+1)^2 + n^2 = 4$

and the fact that $1,2,3,4$ have a representation:

$1 = + 1^2$ $2 = - 1^2 - 2^2 - 3^2 + 4^2$ $3 = -1^2 + 2^2$ $4 = -1^2 - 2^2 + 3^2$

To get one representation of $4m+r$, we inductively get one representation for $4(m-1) + r$, and use the above identity.

Now, as Andre pointed out, given one representation we can extend that to infinitely many representations by writing $0+0+0 \dots$ as $(4 - 4) + (4-4) + (4-4) \dots$ and using the above identity multiple times.