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You’ve just started number theory, either as a university course or because you want to do well in math contests. The first chapters look easy, since everything is about whole numbers you’ve used all your life. Then the problems ask you to prove things about every prime, and the easy feeling disappears.
The book you start with decides whether that moment feels like a puzzle or a wall. The best number theory books for beginners let you experiment with small numbers first and prove things second. A book that starts with formal proofs can make the most playful subject in mathematics feel dry within a few weeks.
My advice is simple. If you’re new to number theory, I’d start with one book: Joseph Silverman’s A Friendly Introduction to Number Theory. It lets you discover patterns before it asks you to prove them, and its short chapters reach real applications such as RSA cryptography.
There are 2 exceptions. If your course already names a textbook, I’d start with that one instead. If you’re preparing for school math contests such as IOQM or RMO, Art of Problem Solving’s Introduction to Number Theory is the better first book.
I studied mathematics through an M.Sc. and taught algebra and exam strategy to NDA and competitive-exam students, where remainder and divisibility questions come up again and again. I haven’t worked through all 10 of these books cover to cover. For the ones I know less closely, my verdict comes from their contents, their exercises and the course each one is built for.
Quick Comparisons
My suggestion is to find your situation in the table and jump straight to that book. The last column tells you how each book teaches, and the free texts after the list cover a full first course.
| Book | Best for | You should know | Style |
|---|---|---|---|
| A Friendly Introduction to Number Theory (Silverman) | Your first number theory book | Basic proofs help | Discovery first, then proofs |
| Introduction to Number Theory (Art of Problem Solving) | School math contests | Strong school algebra | Contest problems with solutions |
| Elementary Number Theory (Burton) | A standard university course | Basic proofs help | Standard textbook |
| Elementary Number Theory (Dudley) | A cheap printed textbook | Basic proofs help | Short and cheap |
| An Introduction to the Theory of Numbers (Hardy and Wright) | A classic reference | Comfort with proofs | Reference, no exercises |
| An Illustrated Theory of Numbers (Weissman) | Learning through diagrams | Undergraduate proofs | Diagrams |
| Introduction to Analytic Number Theory (Apostol) | Analytic methods and primes | Calculus and proofs | Analytic methods |
| An Introduction to the Theory of Numbers (Niven, Zuckerman, Montgomery) | A broad traditional course | Undergraduate proofs | Dense and broad |
| Number Theory: A Historical Approach (Watkins) | Learning through history | Some mathematical maturity | History-led |
| Number Theory and Its History (Ore) | History on a budget | School algebra | History-led, older style |
Best Number Theory Books by Goal
Each heading names the job that book does best, so you can jump straight to yours. Every entry also says who I’d send elsewhere.
Best first book: A Friendly Introduction to Number Theory
Joseph Silverman’s A Friendly Introduction to Number Theory is the book I’d give anyone starting number theory. It starts with Pythagorean triples and small numerical experiments, so you spot a pattern yourself before the book asks you to prove it.
The chapters are short, and the later ones reach RSA cryptography and elliptic curves, which shows a first-time student where the subject leads. That mix of play and proof is what makes it work for beginners.
The 4th edition is current, and the copy linked below is a low-cost Pearson reprint. It is my pick for a first university course and for self-study. If you want results without working the exercises, no book on this list will help you much, and this one least of all.
A Friendly Introduction to Number Theory
- By Joseph H. Silverman
- Best for: a first proof-oriented course
- Level: Proof basics helpful
- Main strength: Uses approachable problems and examples to introduce congruences, Diophantine equations, primes, and cryptography.
- Watch-out: Readers still need patience with proof and exercises.
- Verify the exact edition, format, and seller
Best for olympiad beginners: Art of Problem Solving Introduction to Number Theory
Mathew Crawford’s Introduction to Number Theory from Art of Problem Solving is where I’d send a school student preparing for math contests. It teaches divisibility, modular arithmetic and number bases through problems, and the bundle linked below includes the full solutions manual.
It is written for strong middle and high school students, so it puts less weight on the formal proofs a university course grades you on. I’d pick it for IOQM, RMO or AMC preparation and choose Silverman for a college course.
Once it feels easy, the natural next step is olympiad-level material such as Number Theory: Concepts and Problems by Titu Andreescu and his coauthors.
Art of Problem Solving Introduction to Number Theory
- By Mathew Crawford
- Best for: contest-focused students
- Level: Strong prealgebra and algebra
- Main strength: Builds technique through challenging problems and a separate solutions manual.
- Watch-out: It emphasizes problem solving more than a standard university syllabus.
- Verify the exact edition, format, and seller
Best course textbook: Elementary Number Theory by David Burton
David Burton’s Elementary Number Theory is a standard undergraduate textbook, with a conventional order of topics, many exercises and short historical notes on the mathematicians behind each result.
It is less playful than Silverman, and it is common in university syllabi, including many in India. My recommendation is to buy it when your course assigns it. For self-study, Silverman explains the same first course more warmly.
The 7th edition linked below is expensive new, so it is wise to look for a used or older copy unless your course assigns numbered exercises from it.
- By David M. Burton
- Best for: a structured undergraduate course
- Level: Proof basics helpful
- Main strength: Provides a conventional progression, many exercises, and broad elementary coverage.
- Watch-out: The textbook style is less playful than discovery-led options.
- Verify the exact edition, format, and seller
Best budget textbook: Elementary Number Theory by Underwood Dudley
Underwood Dudley’s Elementary Number Theory is a short, readable Dover edition that covers the classic first-course topics for the price of a paperback novel.
It has none of the extras a modern course package offers, so you get the mathematics and the problems and little else. I’d pick it when money is tight and you want a printed book over a free PDF.
- By Underwood Dudley
- Best for: independent learners
- Level: Proof basics helpful
- Main strength: Offers a readable Dover edition with classic elementary topics at a modest price.
- Watch-out: It is shorter and less supported than modern course packages.
- Verify the exact edition, format, and seller
Best classic reference: An Introduction to the Theory of Numbers by Hardy and Wright
G. H. Hardy and E. M. Wright’s An Introduction to the Theory of Numbers is the classic reference, now in a 6th edition revised by Roger Heath-Brown and Joseph Silverman. It covers far more ground than any first course.
The book has no exercises, which is the clearest sign of its job. I’d keep it as a reference to read around a topic once you know the basics, never as the book you learn from first.
An Introduction to the Theory of Numbers
- By G. H. Hardy and E. M. Wright
- Best for: advanced undergraduates
- Level: Strong proof background
- Main strength: Provides deep classical coverage and long-term reference value.
- Watch-out: It is not the easiest first book and expects mathematical maturity.
- Verify the exact edition, format, and seller
Best for visual learners: An Illustrated Theory of Numbers
Martin Weissman’s An Illustrated Theory of Numbers uses hundreds of diagrams to make primes, congruences and quadratic reciprocity visible. The pictures carry real mathematics and do far more than decorate the page.
Its visual route doesn’t follow every standard course sequence. My suggestion is to choose it if you understand ideas best by seeing them, and to read it beside your course text if the order differs.
An Illustrated Theory of Numbers
- By Martin H. Weissman
- Best for: students who learn number theory through diagrams and patterns
- Level: Proof-oriented undergraduates
- Main strength: Uses rich illustrations to make primes, congruences, reciprocity, and algebraic ideas visible.
- Watch-out: The visual route is substantial but does not mirror every standard course sequence.
- Verify the exact edition, format, and seller
Best for analytic number theory: Introduction to Analytic Number Theory
Tom Apostol’s Introduction to Analytic Number Theory is the standard first book on the analytic side of the subject. It develops arithmetic functions, Dirichlet’s theorem on primes in arithmetic progressions and the prime number theorem with clean proofs and strong exercises.
It assumes calculus, real analysis basics and comfort with proofs. I’d read it after a first course, when you want to know how analysis answers questions about primes.
Introduction to Analytic Number Theory
- By Tom M. Apostol
- Best for: advanced undergraduates studying arithmetic functions and primes
- Level: Calculus and proof experience
- Main strength: Develops classical analytic tools with a clear theorem-proof structure and strong exercises.
- Watch-out: It is not a beginner text and expects mathematical maturity.
- Verify the exact edition, format, and seller
Best traditional textbook: An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery
Ivan Niven, Herbert Zuckerman and Hugh Montgomery’s An Introduction to the Theory of Numbers is a broad traditional textbook covering divisibility, congruences, Diophantine equations and continued fractions.
The presentation is denser than Silverman or Burton. It is my pick for a strong student who wants one durable book that covers a full course and then some.
An Introduction to the Theory of Numbers
- By Ivan Niven , Herbert S. Zuckerman , et al.
- Best for: students who want a durable traditional reference
- Level: Undergraduate proofs
- Main strength: Covers divisibility, congruences, Diophantine equations, continued fractions, and related topics.
- Watch-out: The presentation is denser and older than modern first-course books.
- Verify the exact edition, format, and seller
Best historical course: Number Theory: A Historical Approach
John Watkins’s Number Theory: A Historical Approach follows the subject through the people and problems that shaped it, with real mathematics at every step.
History sets the order, so it is slower for syllabus-based exam preparation. I’d pick it if you understand ideas better once you know why someone needed them.
Number Theory: A Historical Approach
- By John J. Watkins
- Best for: readers who understand ideas better through their origins
- Level: General mathematical maturity
- Main strength: Connects major number-theory problems to the mathematicians and methods that developed them.
- Watch-out: History drives the sequence, so it is less efficient for syllabus-based exam preparation.
- Verify the exact edition, format, and seller
Best history on a budget: Number Theory and Its History
Oystein Ore’s Number Theory and Its History is a much older Dover classic that pairs elementary number theory with its history at a very low price.
Its notation and emphasis show its age. I’d pick it over Watkins when you want the historical route on a small budget.
- By Author or maker shown on the exact listing
- Best for: independent learners who want mathematics with historical narrative
- Level: Algebra and proof curiosity
- Main strength: Pairs classical number theory with the stories and problems that shaped it.
- Watch-out: Its notation and emphasis reflect an older mathematical style.
- Verify the exact edition, format, and seller
Free Number Theory Books and Courses
If I had no budget, I’d pick one free book: Karl-Dieter Crisman’s Number Theory: In Context and Interactive. It is a complete first course with exercises you can also explore by computer. The others below are alternatives for a specific need.
- Number Theory: In Context and Interactive: a free textbook covering modular arithmetic, congruences, primes and quadratic reciprocity, with SageMath cells for experiments.
- Elementary Number Theory: Primes, Congruences, and Secrets: William Stein’s free book, which connects primes and congruences to cryptography and elliptic curves.
- MIT OpenCourseWare 18.781 Theory of Numbers: free lecture notes, problem sets and exams from an MIT undergraduate course.
A free text with exercises is enough for a first course. What a paid copy usually adds is print comfort and a more polished sequence.
How to Choose a Number Theory Book
I’d start by deciding whether you’re studying for a university course or for contests, and then check how much proof experience you have. Those 2 questions settle the choice faster than any bestseller rank.
Contest number theory and university number theory share the same integers. They ask very different questions about them.
Course or Contest?
A university course grades you on proofs, written in full sentences, about general results. A math contest gives you 1 hard problem and rewards the trick that cracks it, often under time pressure.
The topics overlap, but the practice doesn’t. I recommend choosing the book built for the questions you’ll actually be asked: Silverman or Burton for a course, Art of Problem Solving for contests.
Have You Written Proofs Before?
Number theory is often the friendliest first proof course, because you can test every claim on small numbers. If proofs are new to you, Silverman is the safer start, since it lets you check patterns before you prove them.
Writing a proof that someone else can follow is its own skill. I wrote How to Write Mathematics for this stage, and it covers how to set out a readable proof and a clean problem-set solution.
Is the Book Set by Your Course?
If your course names a book, I recommend buying that book, including the edition. Exercise numbers shift between editions, and a mismatched copy turns every assignment into a search.
In India, many B.Sc. and M.Sc. syllabi list Burton or Niven, Zuckerman and Montgomery. If your exam follows one of those, my suggestion is to read the listed book for its notation and keep Silverman beside it for the explanations.
Books I’d Skip as a First Book
Some famous number theory books are excellent and still the wrong place to start. These are the ones I’d keep for later:
- Hardy and Wright: a brilliant reference with no exercises, written for readers who already know the basics.
- Apostol’s Introduction to Analytic Number Theory: the right second book, once calculus, analysis and a first course are behind you.
- Ireland and Rosen’s A Classical Introduction to Modern Number Theory: a graduate-level bridge to algebraic number theory that assumes abstract algebra.
- Olympiad problem collections as a first book: they assume the basics and teach almost nothing from scratch.
- A new edition at full price: if your course doesn’t assign numbered exercises from it, an older Burton teaches the same core for much less.
How I’d Study Number Theory
One main book plus one free resource is enough, and it is wise to finish that book before opening the next. Reading 3 introductions at once feels productive, but it delays the part that actually teaches you, which is solving problems.
The habit I’d build from day one is testing every claim on small numbers before you try to prove it. Checking a congruence for n equal to 1 through 10 often shows you the proof, and it catches a false statement in a minute.
For a University Course
This is the order I’d follow:
- Silverman, or your course’s assigned textbook, one chapter at a time.
- Every exercise tried with small numbers first, then proved.
- Crisman’s free book, only when you want a second explanation of a topic.
- A weekly redo of the proofs you got wrong, in place of reading new pages.
For Math Contests Like IOQM and RMO
For contest preparation, I recommend this order:
- Art of Problem Solving’s Introduction to Number Theory, with every problem attempted before you check the solutions manual.
- Past IOQM and RMO number theory problems, timed, once the book feels comfortable.
- An olympiad-level book such as Number Theory: Concepts and Problems, when the past papers start feeling routine.
For a Second Course
For going deeper after a first course, this is the order that makes sense to me:
- Apostol’s Introduction to Analytic Number Theory, if primes and analysis interest you most.
- Ireland and Rosen in place of Apostol, if you’re drawn to the algebraic side and have taken abstract algebra.
- Hardy and Wright as the reference to read around both.
Number theory connects to the courses around it. For the algebra that grows out of it, I’d look at my abstract algebra book picks. If you’re coming from computer science, my discrete math books cover the number theory that shows up in algorithms and cryptography.
FAQs on Number Theory Books
What are the best number theory books?
My top pick for a first course is Joseph Silverman’s A Friendly Introduction to Number Theory. Art of Problem Solving’s Introduction to Number Theory is the best start for school math contests, and Burton’s Elementary Number Theory is the standard course textbook. Hardy and Wright is the classic reference once you know the basics.
Which number theory book is best for beginners?
I’d start with Silverman’s A Friendly Introduction to Number Theory, because it lets you discover patterns before it asks for proofs. If you want a free alternative, Karl-Dieter Crisman’s Number Theory: In Context and Interactive covers a full first course online.
Which number theory book is best for olympiad preparation?
For a student starting out, my recommendation is Art of Problem Solving’s Introduction to Number Theory, with its solutions manual. Once that feels easy, olympiad-level books such as Number Theory: Concepts and Problems by Andreescu and coauthors are the next step.
Is Hardy and Wright good for beginners?
Not as a first book. It is a classic reference with no exercises, so it is hard to learn from on your own. I’d start with Silverman and read Hardy and Wright later, around the topics you already know.
Are there free number theory books?
Yes. Karl-Dieter Crisman’s Number Theory: In Context and Interactive and William Stein’s Elementary Number Theory: Primes, Congruences, and Secrets are both free online. MIT OpenCourseWare’s 18.781 Theory of Numbers adds free lecture notes and problem sets.
Do I need abstract algebra before number theory?
No. A first number theory course needs only school algebra and some comfort with proofs. Number theory is often a gentler first proof course than abstract algebra, and it makes groups and rings easier to understand later.
Can I learn number theory on my own?
Yes, it is one of the friendliest subjects for self-study. I’d use Silverman and work every exercise with small numbers before trying a proof. When a proof feels shaky, posting it on Math Stack Exchange is a good way to get it checked.
Final Remarks
If your course starts this week, I’d get Silverman and work through the first chapter’s exercises before your first class. If you’re preparing for contests, the Art of Problem Solving book comes first.
Either way, my advice is to buy one book and finish it. Number theory rewards the student who tries 20 small cases before writing a proof, and that habit costs nothing.
Best of luck.
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