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References E.2 General References

There are many good introductory number theory texts.
Gareth A. and J. Mary Jones, Elementary Number Theory, Springer, London, (2005). (Website 1 )
A good introduction with an emphasis on groups, containing interleaved exercises with full answers.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford, (1979) (Website 2  for expanded sixth edition)
A highly regarded text with copious notes, but sometimes more than a little hard to parse with its consecutively numbered theorems and very dense prose.
William Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer, (2008) (Website 3 )
Freely available and the first Sage-enabled number theory text, by the founder of Sage (a number theorist).
Ken Rosen, Elementary Number Theory and its Applications, Pearson, (2011). (Website 4 )
A venerable text with programming exercises that still wear well.
David C. Marshall, Edward Odell, Michael Starbird, Number Theory through Inquiry, Mathematical Association of America, Washington, (2007). (Website 5 )
The topics are very standard, but the approach is quite different; no proofs, only statements. This turns out to be a highly effective pedagogy; see the Academy of Inquiry Based Learning 6  for more information.
R. P. Burn, A pathway into number theory, Cambridge, (1996) (Website 7 )
A very fun inquiry-driven text before there were such things, with a lot of extremely good examples, especially in things like quadratic forms.
John Stillwell, Elements of Number Theory, Springer, (2003) (Website 8 )
More algebraically oriented, with good material on the Pell equation and Gaussian integers – noteworthy for a good treatment of Conway’s river concepts.
Harold Shapiro, Introduction to the Theory of Numbers, Dover, (2008) (No website)
Incredibly comprehensive, at a fairly high level. Good material on averages and odd perfection, immense bibliography and notes in style of [E.2.2], and also inquiry-driven “do-it-yourself” sections. Appears to be out of print.
Anthony Gioia, The Theory of Numbers, Dover, (2001) (No website)
Surprisingly detailed and high-level but has good coverage of several unusual topics such as geometry of numbers.
Marty Erickson, Anthony Vazzana, David Garth, Introduction to Number Theory, second edition, CRC, (2016). (Website 9 )
Enough material for two courses, some fairly advanced, and newly endowed with downloadable Sage worksheets for use with local or online CoCalc 10 .
George Andrews, Number Theory, Dover, (1994) (Website 11 )
Yet another nice reprint from Dover, this one with (as one would expect of the author) great combinatorial content.
H. M. Edwards, Higher Arithmetic: An Algorithmic Introduction to Number Theory, American Mathematical Society, (2006) (Website 12 )
Not so algorithmic, but very, very concrete and constructive. Squares are \(\square\)s, which grows on the reader.
Neville Robbins, Beginning Number Theory, Jones and Bartlett, (2006) (No website)
An out-of-print standard text with many similar topics and interesting historical comments.
Oystein Ore, Invitation to Number Theory, Mathematical Association of America, (1967) (Website 13 )
An older text that is still worth the conversational tone.
Duff Campbell, An Open Door to Number Theory, American Mathematical Society/MAA Press, (2018), (Website 14 )
Careful emphasis throughout on getting a novice student ready for abstract algebra/algebraic number theory, with \(\mathbb{Q}[\sqrt{d}]\) coherent in an elementary text. Don’t miss continued fractions in the service of the Bezout identity and the many interesting projects, including one on the \(p\)-adic numbers.
Róbert Freud and Edit Gyarmati, Number Theory, American Mathematical Society, (2020), (Website 15 )
See my review for MAA reviews 16  of this relatively ambitious text. Could be very interesting to use for a two-semester algebra sequence that starts with number theory.
Cam McLeman, Erin McNicholas, and Colin Starr, Explorations in Number Theory: Commuting through the Numberverse, Springer, (2022), (Website 17 )
An inquiry-friendly introduction with a uniquely elementary tilt toward algebraic number theory. And many, many puns.