### Statement:

Take any whole number

kthat is greater than or equal to 2. Show that there is some numbers(that is allowed to depend onk) so that every positive whole number is a sum ofsk^{th}powers.

### Proof

This problem was first solved by David Hilbert, and was then later proved by G.H. Hardy and John Littlewood in a different way. In this case, the second proof actually gave a lot more insight to the problem. To solve Waring’s problem as it has been stated you don’t need to give an *s*, you only need to show that one exists. Finding the smallest *s* that works is a much more challenging (although arguably more interesting) problem. Using the, now named, *Hardy-Littlewood* *circle method*, Hardy and Littlewood wrote down an expression that approximated the number of ways to write *N* as a sum of *s* *k*^{th} powers:

The proof of Waring’s problem comes from that fact that since this must be positive and also be a whole number, there must be some way to write *N* as a sum of *s* *k*^{th} powers.

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(*Sorry for missing a post last week, my university work is getting a bit hectic so posts may be more sporadic!*)