Problem 3

Find all functions

g:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}

such that

\left(g(m)+n\right)\left(g(n)+m\right)

is a perfect square for all

m,n\in\mathbb{Z}_{>0}

Step 5 of 6: The signs cannot change

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g(k+1)-g(k)=1\text{ for all }k

Detailed analysis

Each difference $g(k+1)-g(k)$ is either $1$ or $-1$ . If two consecutive differences had opposite signs, then either $g(k+2)=g(k)$ or $g(k+2)=g(k)$ , contradicting injectivity. Thus all differences have the same sign. They cannot all be $-1$ , because then $g(k)=g(1)-(k-1)$ would eventually be nonpositive, contrary to $g(k)\in\mathbb{Z}_{>0}$ . Hence $g(k+1)-g(k)=1$ for every $k$ .