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Niedere Zahlentheorie. Teil 1. Teil 2: Additive Zahlentheorie. Reprint. (German) Zbl 0253.10001
Bronx, N. Y.: Chelsea Publishing Company. x, 402 p.; x, 480 p. $ 15.00 (1968).
The two volumes here reprinted together were first published in 1902 [JFM 33.0192.16] and 1910 [JFM 41.0221.10]; they contain many useful references to the classical literature. As one might expect from the title, there is nothing in either volume about the theory of quadratic forms, no use of analytic methods, and very little about algebraic numbers. So the three square theorem has to be stated without proof, but is used e.g. in Wieferich’s proof of the nine cube theorem. Dirichlet’s theorem on primes in an arithmetic progression is also stated without proof.
In the second volume, on additive number theory, the results on Waring’s problem, obtained by elementary arguments based on algebraic identities, are nearly all obsolete now, because of the stronger results obtained analytically by Hardy and Littlewood. In the rest (over 95 per cent) of the book, I do not notice anything else that has been superseded by later work.
Reviewer: G. L. Watson

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11Pxx Additive number theory; partitions
01A75 Collected or selected works; reprintings or translations of classics