Further basic results were proved in the early 20th century.

He also proved similar results for 6 and 8 squares.

A similar result with a higher value of was recently proved by .

The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.

The argument given above shows that the result can be proved without using the axiom of choice.

Thus, many results from ordinary mathematics can be proved in more interesting categories.

The result can also be proved by the technique of reverse reconstruction.

It proved excellent results within five years, and every other state took it as role model.

It is useful for proving results about the normal order of an arithmetic function.

No such result has been proved in general for an arbitrary but reasonable model of computation.