Video2013
Infinite Primes
Numberphile
Presents Euclid's proof that the primes do not run out, and uses it to show what a proof by contradiction actually establishes as opposed to what checking many cases can.
link checked 17 Sept 2026FreeEntry
Number Theory
The behaviour of the integers — the oldest questions in mathematics and several of the hardest.
13 topics · 14 curated works
Topics
- 01Foundations & Overviews
- 02Elementary Number Theory1
- 03Analytic Number Theory1
- 04Algebraic Number Theory1
- 05Diophantine Equations1
- 06Distribution of Primes2
- 07The Riemann Hypothesis2
- 08Modular Forms1
- 09Elliptic Curves1
- 10p-adic Analysis1
- 11Transcendence Theory1
- 12Additive Combinatorics1
- 13Computational Number Theory1
Reading in Number Theory
14A way in
Start here
No prior grounding assumed.
Infinite Primes
Numberphile · 2013
Presents Euclid's proof that the primes do not run out, and uses it to show what a proof by contradiction actually establishes as opposed to what…
+1 more at this level
Then
Assumes you know the vocabulary.
A Computational Introduction to Number Theory and Algebra
Victor Shoup · 2005
Presents number theory and algebra through the algorithms that make them computable, treating problems like discrete logarithm and integer factoring…
+1 more at this level
Go deeper
Primary sources and full treatments.
On the Number of Primes Less Than a Given Magnitude
Bernhard Riemann · 1859
Connects the distribution of primes to the zeros of a single complex function, introducing the zeta function's analytic continuation and conjecturing…
+9 more at this level
12 of 14 works
Video2016
Visualizing the Riemann zeta function and analytic continuation
Grant Sanderson (3Blue1Brown)
Extends the zeta function beyond its original domain of convergence by visualising analytic continuation directly, arguing this is what actually gives meaning to statements about where its zeros lie.
link checked 17 Sept 2026FreeEntry
Book2005
A Computational Introduction to Number Theory and Algebra
Victor Shoup
Presents number theory and algebra through the algorithms that make them computable, treating problems like discrete logarithm and integer factoring as the load-bearing structure of modern cryptography rather than classical curiosities.
link checked 17 Sept 2026FreeIntermediate
Book2009
Elementary Number Theory: Primes, Congruences, and Secrets
William Stein
Builds elementary number theory around algorithms a computer can actually run, arguing that primality testing, factoring and cryptography are the natural motivation for congruences rather than an afterthought.
link checked 17 Sept 2026FreeIntermediate
Paper1859
On the Number of Primes Less Than a Given Magnitude
Bernhard Riemann
Connects the distribution of primes to the zeros of a single complex function, introducing the zeta function's analytic continuation and conjecturing where its non-trivial zeros lie.
9 pageslink checked 17 Sept 2026FreeAdvanced
Paper1896
Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques
Jacques Hadamard
Proves the Riemann zeta function has no zeros on the line Re(s)=1, the missing step that turns Riemann's 1859 paper into the first complete proof of the Prime Number Theorem.
60 pageslink checked 17 Sept 2026FreeAdvanced
Paper1995
Modular Elliptic Curves and Fermat's Last Theorem
Andrew Wiles
Proves that semistable elliptic curves are modular, closing the gap that (with Ribet's earlier work) shows Fermat's equation has no positive integer solutions for exponents greater than two.
109 pageslink checked 17 Sept 2026FreeAdvanced
Course1996
Algebraic Number Theory
J.S. Milne
Develops rings of integers, Dedekind domains and the finiteness of the class number as the right generalisation of unique factorisation once ordinary integers no longer suffice.
link checked 17 Sept 2026FreeAdvanced
Course1997
Modular Functions and Modular Forms
J.S. Milne
Develops modular curves as moduli spaces for elliptic curves with level structure, arguing the geometric viewpoint explains identities that the classical q-expansion definition leaves mysterious.
link checked 17 Sept 2026FreeAdvanced
Course2000
An Introduction to p-adic Numbers and p-adic Analysis
Andrew Baker
Builds the p-adic numbers from Cauchy sequences under a non-Archimedean norm, showing that many arguments from ordinary analysis carry over once distance is redefined by divisibility rather than size.
link checked 17 Sept 2026FreeAdvanced
Course2007
18.785: Analytic Number Theory
Kiran Kedlaya (MIT)
Develops the machinery of zeta and L-functions to prove the prime number theorem and its refinement to arithmetic progressions, treating analytic continuation as the engine that turns a counting problem into a question about complex zeros.
link checked 17 Sept 2026FreeAdvanced
Course2010
Diophantine Approximation, Irrationality and Transcendence
Michel Waldschmidt
Surveys the methods that prove specific constants irrational or transcendental, arguing that quantifying how well a number resists rational approximation is the thread linking Liouville's theorem to modern transcendence results.
link checked 17 Sept 2026FreeAdvanced
In order written
1859 – 2021- 1859On the Number of Primes Less Than a Given MagnitudeBernhard Riemann
- 1995Modular Elliptic Curves and Fermat's Last TheoremAndrew Wiles
- 1996Algebraic Number TheoryJ.S. Milne
- 1997Modular Functions and Modular FormsJ.S. Milne
- 2000An Introduction to p-adic Numbers and p-adic AnalysisAndrew Baker
- 2005A Computational Introduction to Number Theory and AlgebraVictor Shoup
- 200718.785: Analytic Number TheoryKiran Kedlaya (MIT)
- 2009Elementary Number Theory: Primes, Congruences, and SecretsWilliam Stein
- 2010Diophantine Approximation, Irrationality and TranscendenceMichel Waldschmidt
- 2013Infinite PrimesNumberphile
- 2016Visualizing the Riemann zeta function and analytic continuationGrant Sanderson (3Blue1Brown)
- 2017Graph Theory and Additive CombinatoricsYufei Zhao (MIT)
- 202118.783: Elliptic CurvesAndrew Sutherland (MIT)