Skip to main content
Logo image

Number Theory: In Context and Interactive

Karl-Dieter Crisman

References E.2 General References

There are many good introductory number theory texts.
[1]
Gareth A. and J. Mary Jones, Elementary Number Theory, Springer, London, (2005). (Website
 1 
www.springer.com/us/book/9783540761976
)
Note.
A good introduction with an emphasis on groups, containing interleaved exercises with full answers.
[2]
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford, (1979) (Website
 2 
global.oup.com/academic/product/an-introduction-to-the-theory-of-numbers-9780199219865
for expanded sixth edition)
Note.
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.
[3]
William Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer, (2008) (Website
 3 
wstein.org/ent/
)
Note.
Freely available and the first Sage-enabled number theory text, by the founder of Sage (a number theorist).
[4]
Ken Rosen, Elementary Number Theory and its Applications, Pearson, (2011). (Website
 4 
www.pearsonhighered.com/educator/product/Elementary-Number-Theory/9780321500311.page
)
Note.
A venerable text with programming exercises that still wear well.
[5]
David C. Marshall, Edward Odell, Michael Starbird, Number Theory through Inquiry, Mathematical Association of America, Washington, (2007). (Website
 5 
www.maa.org/publications/ebooks/number-theory-through-inquiry
)
Note.
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 
www.inquirybasedlearning.org
for more information.
[6]
R. P. Burn, A pathway into number theory, Cambridge, (1996) (Website
 7 
www.cambridge.org/us/academic/subjects/mathematics/number-theory/pathway-number-theory-2nd-edition
)
Note.
A very fun inquiry-driven text before there were such things, with a lot of extremely good examples, especially in things like quadratic forms.
[7]
John Stillwell, Elements of Number Theory, Springer, (2003) (Website
 8 
www.springer.com/us/book/9780387955872
)
Note.
More algebraically oriented, with good material on the Pell equation and Gaussian integers – noteworthy for a good treatment of Conway’s river concepts.
[8]
Harold Shapiro, Introduction to the Theory of Numbers, Dover, (2008) (No website)
Note.
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.
[9]
Anthony Gioia, The Theory of Numbers, Dover, (2001) (No website)
Note.
Surprisingly detailed and high-level but has good coverage of several unusual topics such as geometry of numbers.
[10]
Marty Erickson, Anthony Vazzana, David Garth, Introduction to Number Theory, second edition, CRC, (2016). (Website
 9 
tvazzana.sites.truman.edu/introduction-to-number-theory/
)
Note.
Enough material for two courses, some fairly advanced, and newly endowed with downloadable Sage worksheets for use with local or online CoCalc
 10 
cocalc.com
.
[11]
George Andrews, Number Theory, Dover, (1994) (Website
 11 
store.doverpublications.com/0486682528.html
)
Note.
Yet another nice reprint from Dover, this one with (as one would expect of the author) great combinatorial content.
[12]
H. M. Edwards, Higher Arithmetic: An Algorithmic Introduction to Number Theory, American Mathematical Society, (2006) (Website
 12 
bookstore.ams.org/stml-45/
)
Note.
Not so algorithmic, but very, very concrete and constructive. Squares are \(\square\)s, which grows on the reader.
[13]
Neville Robbins, Beginning Number Theory, Jones and Bartlett, (2006) (No website)
Note.
An out-of-print standard text with many similar topics and interesting historical comments.
[14]
Oystein Ore, Invitation to Number Theory, Mathematical Association of America, (1967) (Website
 13 
www.maa.org/press/ebooks/invitation-to-number-theory
)
Note.
An older text that is still worth the conversational tone.
[15]
Duff Campbell, An Open Door to Number Theory, American Mathematical Society/MAA Press, (2018), (Website
 14 
bookstore.ams.org/text-39
)
Note.
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.
[16]
Róbert Freud and Edit Gyarmati, Number Theory, American Mathematical Society, (2020), (Website
 15 
bookstore.ams.org/amstext-48/
)
Note.
See my review for MAA reviews
 16 
www.maa.org/press/maa-reviews/number-theory-2
of this relatively ambitious text. Could be very interesting to use for a two-semester algebra sequence that starts with number theory.
[17]
Cam McLeman, Erin McNicholas, and Colin Starr, Explorations in Number Theory: Commuting through the Numberverse, Springer, (2022), (Website
 17 
link.springer.com/book/10.1007/978-3-030-98931-6
)
Note.
An inquiry-friendly introduction with a uniquely elementary tilt toward algebraic number theory. And many, many puns.
PrevTopNext