Take any whole number k that is greater than or equal to 2. Show that there is some number s (that is allowed to depend on k) so that every positive whole number is a sum of s kth powers.
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 kth 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 kth 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!)