non-abelian simple groups then known having order less than 1 billion. He listed 53 of the 56 having order less than 1 million. The remaining three were found in 1960, 1965, and 1967. Dickson worked on finite fields and extended the theory of linear associative algebras initiated by Joseph Wedderburn and Cartan. He started the study of modular invariants of a group. In 1905, Wedderburn, then at Chicago on a Carnegie Fellowship, published a paper that included three claimed proofs of a theorem stating that all finite division algebras were commutative, now known as Wedderburn's theorem. The proofs all made clever use of the interplay between the additive group of a finite division algebra A, and the multiplicative group A* = A − {0}. Karen Parshall noted that the first of these three proofs had a gap not noticed at the time. Dickson also found a proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority. But Dickson also noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. She concluded that Dickson should be credited with the first correct proof. Dickson's search for a counterexample to Wedderburn's theorem led him to investigate nonassociative algebras, and in a series of papers he found all possible three and four-dimensional (nonassociative) division algebras over a field. In 1919 Dickson constructed Cayley numbers by a doubling process starting with quaternions . His method was extended to a doubling of to produce , and of to produce by A. A. Albert in 1922, and the procedure is known now as the Cayley–Dickson construction of composition algebras. The number theorist Dickson proved many interesting results in number theory, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory. He proved the Waring's problem for under the further condition of independently of Subbayya Sivasankaranarayana Pillai who proved it for ahead of him. The three-volume History of the Theory of Numbers (1919–23) is still much consulted today, covering divisibility and primality, Diophantine analysis, and quadratic and higher forms. The work contains little interpretation and makes no attempt to contextualize the results being described, yet it contains essentially every significant number theoretic idea from the dawn of mathematics up to the 1920s except for quadratic reciprocity and higher reciprocity laws. A planned fourth volume on these topics was never written.