The larger portion of the theorems in Diophantine Analysis probably existed first as empirical or conjectural theorems. Many of them passed to the state of proved theorems before they left the hands of those who discovered them; many others were proved in the same generation in which they were made public; not a few required a longer period for their proof; and several remain today as a silent challenge to the skill and power of contemporary mathematicians. The remarks may be illustrated with a brief account of the history of the problem of representing numbers (that is, positive integers) as sums of squares of integers and of higher powers. Anyone interested in further details will find them in the comprehensive account of Diophantine Analysis which fills volume II (xxvi + 803 pages) of L. E. Dickson's “History of the Theory of Numbers,” Carnegie Institution, Washington, D. C. We shall make free use of the material summarized in a masterly way in this volume.
Dickson's history of the theory of numbers