Goutham wrote: Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying \[m(m + k) = n(n + 1).\] If $k=1$, $(m,n)=(1,1)$ is a solution and so $k=1$ fits. If $k=2$, the equation $m(m+2)=n(n+1)$ may be written $(2m-2n+1)(2m+2n+3)=3$ which is impossible since $2m+2n+3\ge 7$ If $k=3$, the equation $m(m+3)=n(n+1)$ may be written $(2m-2n+2)(2m+2n+4)=8$ which is impossible since $2m+2n+4\ge 8$ and $2m-2n+2\ne 1$ If $k=2p\ge 4$, then $(m,n)=(p^2-p,p^2-1)$ fits If $k=2p+1\ge 5$, then $(m,n)=(\frac{p(p-1)}2,\frac{p(p+1)}2-1)$ fits Hence the answer : $\boxed{\mathbb N\setminus\{2,3\}}$