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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!

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Because a square is both a polygonal number and a perfect power, Lagrange’s four-square theorem can be generalized/extended in several ways.

First, it is the case $n=4$ of the Fermat polygonal number theorem, first stated without proof by Fermat (in 1638), and first proven by Cauchy (in 1813):

Theorem: Every positive integer is a sum of at most $n$ $n$-gonal numbers — i.e., every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on.

That theorem in turn can likely be generalized/extended to figurate numbers, as conjectured by Pollock (in 1850).

The second obvious way to generalize/extend Lagrange’s theorem is the one you’ve described, known as Waring’s Problem, first proposed by Waring (in 1770).

Question: For each natural number $k$, is there an associated positive integer $s$ such that every natural number is the sum of the $k$th powers of at most $s$ natural numbers?

It is currently an open area of research in additive number theory.

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