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I am looking for a good textbook suitable for graduate or advanced undergraduate students who want to explore algorithmic number theory. Specifically, algorithms for primality testing, and factoring composite numbers. Can anyone suggest a good text book for this purpose? Thank you in advance.

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Henri Cohen's books ? – Chandan Singh Dalawat Jun 9 '13 at 14:56
Definitely Henri Cohen's "A course in computational algebraic number theory" is the best place to start. At a lower level David M. Bressoud "Factorization and Primality Testing" is a nice introduction. – Chris Wuthrich Jun 9 '13 at 15:12
What do you mean by "explore". Should they learn the subject seriously or just get a bit of a vague idea. For the former I strongly endorse the suggestion of Henri Cohen's book "A course in computational algebraic number theory" (while it says 'algebraic' in the title also classical primetesting algos that have not much to do with alg. numb. theory are covered). For the latter with some guidance it could also work well; but then I think one should not read it in linear order. – user9072 Jun 9 '13 at 15:14
Two texts that would support such study, if not quite graduate level, are Hans Riesel's book on factorization and primality testing, and Crandall and Pomerance's book focusing on prime numbers. Mastering the contents of both books puts you well on the road to such study. Gerhard "Might Remember The Titles Someday" Paseman, 2013.06.09 – Gerhard Paseman Jun 9 '13 at 16:01
William Stein has a book about doing number theory with SAGE. – S. Carnahan Jun 10 '13 at 13:42

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