Waring's Problem
For each integer
\[k \ge 2\]
there exists a positive integer \[g(k)\]
such that every positive integer can be expressed as a sum of at most \[g(k)\]
kth powers.Lagrange's Four Square Theorem confirms that
\[g(2)=4\]
. Waring claimed that \[g(3)=9\]
and \[g(4)=19\]
and these have been confirmed.\[23=1^3+1^3+1^3+1^3+1^3+1^3+1^3+2^3+2^3\]
\[239=1^3+1^3+1^3+3^3+3^3+3^3+3^3+4^3+4^3\]
All integers greater than 239 can be written as a sum of at most 8 cubes and it is known that only finitely many require 9 cubes, so that from some point only seven will be enough.
A lower limit
\[g(k) \ge int ((\frac{3}{2})^k)+2^k-2\]
was discovered in 1772, In fact this expression give that value of \[g(k)\]
for all \[k\]
yet verified and probably holds for all \[k\]
. 
