Find all pairs of positive integers $(n,k)$

Question

Find all positive integers $n$ and $k$ such that: $n^3-5n+10=2^k$

What I found is: by using $\bmod 7$ I proved that $3$ has to divide $k$ and it's easy to show that $v_2(n)=1$. Further more, $n \equiv 2$ or $n \equiv 3 \bmod 7$

Answer 1

Well, if $k$ is divisible by 3, then $2^k$ is a cube. How close can it get to another cube, which is $n^3$? Not too close, I guess. So in short, it is one of the following:

• $2^k=n^3,\;5n-10=0,\;n=2$;
• $2^k=(n-a)^3,\;a\ge1$, which means that $-5n+10\le-3n^2+3n-1$
• $2^k=(n+a)^3,\;a\ge1$, which means that $-5n+10\ge3n^2+3n+1$

The latter two options essentially boil down to a finite check.