Prove: There is no positive integer n such that n^2+n^3=100.

Both functions $f1(n)=n^2$ and $f2(n)=n^3$ is increasing if n is positive integer, which means their sum is increasing too. 5^3=125, so, the easiest way to prove the statement is to calculate $f(n)=n^2+n^3$ for integers from 1 to 4.

n=1: $f(1) = 2$

n=2:$f(2) = 12$

n=3:$f(3) = 36$

n=4:$f(4) = 80$

If n is negative integer, then n^2 + n^3 ≤ 0, so it cannot be equal to 100.

If n = 0 then f(0) = 0.

The statement has been proven.