In 1770 it was proven that every positive integer is a sum of at most four perfect squares (and no fewer suffices). In 1912 it was proven that every positive integer is a sum of at most 9 positive cubes (and no fewer suffices). Then in 1940 it was proven for 73 sixth-powers; then in 1964 for 36 positive fifth-powers; then in 1986 for 19 fourth-powers. A formula for the number of k-th powers, k > 6, was conjectured in 1944, and proven to have at most finitely many exceptions in 1957. As of 1990 it is known the conjecture is sound up until k = 471 million.
What made fourth powers so hard?
Presumably unrelated, but four dimensions are generally weird topologically:
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