This is a straightforward introduction to elementary number theory, that doesn’t require anything beyond high school math and that covers the classical results. The present volume was originally published in 1951 by Wiley and then reprinted by Chelsea in 1964. It is especially strong on Diophantine equations (the author’s research specialty). Another unusual feature is elementary proofs of Dirichlet’s theorem on primes in arithmetic progressions for several special cases. For example, there’s a well-known elementary proof that there are infinitely many primes of the form \(4n - 1\), but this book proves a number of other special cases. It was probably the first elementary book to include Selberg’s elementary proof of the Prime Number Theorem, although it follows Selberg’s original (and difficult) proof very closely. The book also has an enormous number of exercises, many of them difficult (there are no solutions, though).

There were a number of similar textbooks published in the first half of the twentieth century, such as Vinogradov’s *Elements of Number Theory*. Each of them has its specialties and strengths. Since 1950 not much has come out along these lines, although a good modern version is Rassias’s 2011 *Problem-Solving and Selected Topics in Number Theory*. It covers roughly the same ground but has some new proofs; it also has a large collection of new worked problems, many taken from Math Olympiads, journal problem columns, and the Putnam exams.

These books are not as comprehensive as the “big gun” introductions, such as Hardy & Wright’s *An Introduction to the Theory of Numbers*, or Niven & Zuckerman & Montgomery’s *An Introduction to the Theory of Numbers*. They also do not use algebraic language or approaches, as do the more advanced introductions such as Rose’s *A Course in Number Theory* or Ireland & Rosen’s *A Classical Introduction to Modern Number Theory*.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.