So, everyone knows the famous Lagrange's four-square theorem, which states, that every positive integer can be written down as the sum of $4$ square numbers. Since $4=2^2$, and $2$ represents the square numbers, could this be stated for bigger numbers too? For example, $8=2^3$, so we could state, that every positive integer can be written down as the sum of $8$ cube numbers? I tried to find a counter-example for this statement, but didn't have any success.
What do you think about this idea? Can you tell me a counter example or a problem with the thinking? Thanks!