Binomial coefficients are hardly ever powers. That is, there are strong restrictions on when the equation
has integer solutions for l ≥ 2.
- There are infinitely many solutions for k = 1 or 2.
- If k = 3 and l = 2, there is only one solution: n = 50 and m = 140.
- If k is 4 or larger, there are no solutions.
(Binomial coefficients remain unchanged when you replace k with n-k. So when we say there are no solutions for k ≥ 4, we also assume n-k ≥ 4.)
Source: Proofs from the Book