A few months ago, I gave a talk at the IMO in Bremen on “Structure and randomness in the prime numbers”. I have now converted the slides from that talk into a more traditional paper (7 pages in length), for submission to a Festschrift for the Bremen Olympiad. The content is much the same as the slides, but some references have been added.
Recent Comments
 | Island of Blue-Eyed… on The blue-eyed islanders puzzle |
| Island of Blue-Eyed… on About |
 | Anonymous on Equidistribution of Syracuse r… |
| Anonymous on Equidistribution of Syracuse r… |
| Anonymous on Equidistribution of Syracuse r… |
 | Li Jiang on Equidistribution of Syracuse r… |
| Li Jiang on Equidistribution of Syracuse r… |
| Li Jiang on Equidistribution of Syracuse r… |
| Li Jiang on Equidistribution of Syracuse r… |
| Anonymous on The Ionescu-Wainger multiplier… |
| Anonymous on Equidistribution of Syracuse r… |
 | The Ionescu-Wainger… on The sunflower lemma via Shanno… |
| The Ionescu-Wainger… on Pointwise ergodic theorems for… |
 | Hollis Williams on Equidistribution of Syracuse r… |
| Li Jiang on Equidistribution of Syracuse r… |
Articles by others
- Gene Weingarten – Pearls before breakfast
- Isaac Asimov – The relativity of wrong
- Jonah Lehrer – Don't! – the secret of self-control
- Julianne Dalcanton – The cult of genius
- Nassim Taleb – The fourth quadrant: a map of the limits of statistics
- Paul Graham – What You'll Wish You'd Known
- Po Bronson – How not to talk to your kids
- Scott Aaronson – Ten signs a claimed mathematical proof is wrong
- Timothy Gowers – Elsevier — my part in its downfall
- Timothy Gowers – The two cultures of mathematics
- William Thurston – On proof and progress in mathematics
Diversions
- Abstruse Goose
- Assembler
- BoxCar2D
- Factcheck.org
- FiveThirtyEight
- Gapminder
- Literally Unbelievable
- Planarity
- PolitiFact
- Quite Interesting
- snopes
- Strange maps
- Television tropes and idioms
- The Economist
- The Onion
- The Straight Dope
- This American Life on the financial crisis I
- This American Life on the financial crisis II
- What if? (xkcd)
- xkcd
Mathematics
- 0xDE
- A Mind for Madness
- A Portion of the Book
- Absolutely useless
- Alex Sisto
- Almost Originality
- AMS blogs
- AMS Graduate Student Blog
- Analysis & PDE
- Analysis & PDE Conferences
- Annoying Precision
- Area 777
- Ars Mathematica
- ATLAS of Finite Group Representations
- Automorphic forum
- Avzel's journal
- Blog on Math Blogs
- blogderbeweise
- Bubbles Bad; Ripples Good
- Cédric Villani
- Climbing Mount Bourbaki
- Coloquio Oleis
- Combinatorics and more
- Compressed sensing resources
- Computational Complexity
- Concrete nonsense
- David Mumford's blog
- Delta epsilons
- DispersiveWiki
- Disquisitiones Mathematicae
- Embûches tissues
- Emmanuel Kowalski’s blog
- Equatorial Mathematics
- fff
- Floer Homology
- Frank Morgan’s blog
- Gérard Besson's Blog
- Gödel’s Lost Letter and P=NP
- Geometric Group Theory
- Geometry and the imagination
- Geometry Bulletin Board
- George Shakan
- Girl's Angle
- God Plays Dice
- Good Math, Bad Math
- Graduated Understanding
- Hydrobates
- I Can't Believe It's Not Random!
- I Woke Up In A Strange Place
- Igor Pak's blog
- Images des mathématiques
- In theory
- James Colliander's Blog
- Jérôme Buzzi’s Mathematical Ramblings
- Joel David Hamkins
- Journal of the American Mathematical Society
- Kill Math
- Le Petit Chercheur Illustré
- Lemma Meringue
- Lewko's blog
- Libres pensées d’un mathématicien ordinaire
- LMS blogs page
- Low Dimensional Topology
- M-Phi
- Mark Sapir's blog
- Math Overflow
- Math3ma
- Mathbabe
- Mathblogging
- Mathematical musings
- Mathematics Illuminated
- Mathematics in Australia
- Mathematics Jobs Wiki
- Mathematics Stack Exchange
- Mathematics under the Microscope
- Mathematics without apologies
- Mathlog
- Mathtube
- Matt Baker's Math Blog
- Mixedmath
- Motivic stuff
- Much ado about nothing
- Multiple Choice Quiz Wiki
- nLab
- Noncommutative geometry blog
- Nonlocal equations wiki
- Nuit-blanche
- Number theory web
- outofprintmath
- Pattern of Ideas
- Pengfei Zhang's blog
- Persiflage
- Peter Cameron's Blog
- Phillipe LeFloch's blog
- ProofWiki
- Quomodocumque
- Ramis Movassagh's blog
- Random Math
- Reasonable Deviations
- Regularize
- Rigorous Trivialities
- Roots of unity
- Secret Blogging Seminar
- Selected Papers Network
- Sergei Denisov's blog
- Short, Fat Matrices
- Shtetl-Optimized
- Shuanglin's Blog
- Since it is not…
- Sketches of topology
- Snapshots in Mathematics !
- Soft questions
- Stacks Project Blog
- SymOmega
- tcs math
- TeX, LaTeX, and friends
- The accidental mathematician
- The Cost of Knowledge
- The Everything Seminar
- The Geomblog
- The L-function and modular forms database
- The n-Category Café
- The n-geometry cafe
- The On-Line Blog of Integer Sequences
- The polylogblog
- The polymath blog
- The polymath wiki
- The Tricki
- The twofold gaze
- The Unapologetic Mathematician
- The value of the variable
- The World Digital Mathematical Library
- Theoretical Computer Science – StackExchange
- Tim Gowers’ blog
- Tim Gowers’ mathematical discussions
- Todd and Vishal’s blog
- Van Vu's blog
- Vaughn Climenhaga
- Vieux Girondin
- Visual Insight
- Vivatsgasse 7
- Williams College Math/Stat Blog
- Windows on Theory
- Wiskundemeisjes
- XOR’s hammer
- Yufei Zhao's blog
- Zhenghe's Blog
Selected articles
- A cheap version of nonstandard analysis
- A review of probability theory
- American Academy of Arts and Sciences speech
- Amplification, arbitrage, and the tensor power trick
- An airport-inspired puzzle
- Benford's law, Zipf's law, and the Pareto distribution
- Compressed sensing and single-pixel cameras
- Einstein’s derivation of E=mc^2
- On multiple choice questions in mathematics
- Problem solving strategies
- Quantum mechanics and Tomb Raider
- Real analysis problem solving strategies
- Sailing into the wind, or faster than the wind
- Simons lectures on structure and randomness
- Small samples, and the margin of error
- Soft analysis, hard analysis, and the finite convergence principle
- The blue-eyed islanders puzzle
- The cosmic distance ladder
- The federal budget, rescaled
- Ultrafilters, non-standard analysis, and epsilon management
- What is a gauge?
- What is good mathematics?
- Why global regularity for Navier-Stokes is hard
Software
The sciences
Top Posts
- The Ionescu-Wainger multiplier theorem and the adeles
- Career advice
- Pointwise ergodic theorems for non-conventional bilinear polynomial averages
- Does one have to be a genius to do maths?
- Equidistribution of Syracuse random variables and density of Collatz preimages
- On writing
- Books
- About
- There’s more to mathematics than rigour and proofs
- The sunflower lemma via Shannon entropy
Archives
- August 2020 (2)
- July 2020 (2)
- June 2020 (1)
- May 2020 (2)
- April 2020 (3)
- March 2020 (9)
- February 2020 (1)
- January 2020 (3)
- December 2019 (4)
- November 2019 (2)
- September 2019 (2)
- August 2019 (3)
- July 2019 (2)
- June 2019 (4)
- May 2019 (6)
- April 2019 (4)
- March 2019 (2)
- February 2019 (5)
- January 2019 (1)
- December 2018 (6)
- November 2018 (2)
- October 2018 (2)
- September 2018 (5)
- August 2018 (3)
- July 2018 (3)
- June 2018 (1)
- May 2018 (4)
- April 2018 (4)
- March 2018 (5)
- February 2018 (4)
- January 2018 (5)
- December 2017 (5)
- November 2017 (3)
- October 2017 (4)
- September 2017 (4)
- August 2017 (5)
- July 2017 (5)
- June 2017 (1)
- May 2017 (3)
- April 2017 (2)
- March 2017 (3)
- February 2017 (1)
- January 2017 (2)
- December 2016 (2)
- November 2016 (2)
- October 2016 (5)
- September 2016 (4)
- August 2016 (4)
- July 2016 (1)
- June 2016 (3)
- May 2016 (5)
- April 2016 (2)
- March 2016 (6)
- February 2016 (2)
- January 2016 (1)
- December 2015 (4)
- November 2015 (6)
- October 2015 (5)
- September 2015 (5)
- August 2015 (4)
- July 2015 (7)
- June 2015 (1)
- May 2015 (5)
- April 2015 (4)
- March 2015 (3)
- February 2015 (4)
- January 2015 (4)
- December 2014 (6)
- November 2014 (5)
- October 2014 (4)
- September 2014 (3)
- August 2014 (4)
- July 2014 (5)
- June 2014 (5)
- May 2014 (5)
- April 2014 (2)
- March 2014 (4)
- February 2014 (5)
- January 2014 (4)
- December 2013 (4)
- November 2013 (5)
- October 2013 (4)
- September 2013 (5)
- August 2013 (1)
- July 2013 (7)
- June 2013 (12)
- May 2013 (4)
- April 2013 (2)
- March 2013 (2)
- February 2013 (6)
- January 2013 (1)
- December 2012 (4)
- November 2012 (7)
- October 2012 (6)
- September 2012 (4)
- August 2012 (3)
- July 2012 (4)
- June 2012 (3)
- May 2012 (3)
- April 2012 (4)
- March 2012 (5)
- February 2012 (5)
- January 2012 (4)
- December 2011 (8)
- November 2011 (8)
- October 2011 (7)
- September 2011 (6)
- August 2011 (8)
- July 2011 (9)
- June 2011 (8)
- May 2011 (11)
- April 2011 (3)
- March 2011 (10)
- February 2011 (3)
- January 2011 (5)
- December 2010 (5)
- November 2010 (6)
- October 2010 (9)
- September 2010 (9)
- August 2010 (3)
- July 2010 (4)
- June 2010 (8)
- May 2010 (8)
- April 2010 (8)
- March 2010 (8)
- February 2010 (10)
- January 2010 (12)
- December 2009 (11)
- November 2009 (8)
- October 2009 (15)
- September 2009 (6)
- August 2009 (13)
- July 2009 (10)
- June 2009 (11)
- May 2009 (9)
- April 2009 (11)
- March 2009 (14)
- February 2009 (13)
- January 2009 (18)
- December 2008 (8)
- November 2008 (9)
- October 2008 (10)
- September 2008 (5)
- August 2008 (6)
- July 2008 (7)
- June 2008 (8)
- May 2008 (11)
- April 2008 (12)
- March 2008 (12)
- February 2008 (13)
- January 2008 (17)
- December 2007 (10)
- November 2007 (9)
- October 2007 (9)
- September 2007 (7)
- August 2007 (9)
- July 2007 (9)
- June 2007 (6)
- May 2007 (10)
- April 2007 (11)
- March 2007 (9)
- February 2007 (4)
Categories
- expository (285)
- tricks (10)
- guest blog (10)
- Mathematics (795)
- math.AC (8)
- math.AG (41)
- math.AP (111)
- math.AT (17)
- math.CA (170)
- math.CO (174)
- math.CT (7)
- math.CV (28)
- math.DG (37)
- math.DS (82)
- math.FA (24)
- math.GM (12)
- math.GN (21)
- math.GR (86)
- math.GT (16)
- math.HO (12)
- math.IT (13)
- math.LO (49)
- math.MG (43)
- math.MP (28)
- math.NA (23)
- math.NT (177)
- math.OA (19)
- math.PR (99)
- math.QA (6)
- math.RA (39)
- math.RT (21)
- math.SG (4)
- math.SP (47)
- math.ST (8)
- non-technical (158)
- admin (44)
- advertising (39)
- diversions (4)
- media (13)
- journals (3)
- obituary (14)
- opinion (30)
- paper (205)
- question (123)
- polymath (85)
- talk (67)
- DLS (20)
- teaching (181)
- 245A – Real analysis (11)
- 245B – Real analysis (21)
- 245C – Real analysis (6)
- 246A – complex analysis (9)
- 246C – complex analysis (5)
- 247B – Classical Fourier Analysis (5)
- 254A – analytic prime number theory (19)
- 254A – ergodic theory (18)
- 254A – Hilbert's fifth problem (12)
- 254A – Incompressible fluid equations (5)
- 254A – random matrices (14)
- 254B – expansion in groups (8)
- 254B – Higher order Fourier analysis (9)
- 255B – incompressible Euler equations (2)
- 275A – probability theory (6)
- 285G – poincare conjecture (20)
- Logic reading seminar (8)
- travel (26)
additive combinatorics
almost orthogonality
approximate groups
arithmetic progressions
Ben Green
Cauchy-Schwarz
Cayley graphs
central limit theorem
Chowla conjecture
circular law
compactness
compressed sensing
correspondence principle
distributions
eigenvalues
Elias Stein
Emmanuel Breuillard
entropy
equidistribution
ergodic theory
Euler equations
expander graphs
exponential sums
finite fields
Fourier transform
Freiman's theorem
Gowers uniformity norms
graph theory
Gromov's theorem
GUE
Hilbert's fifth problem
incompressible Euler equations
inverse conjecture
Kaisa Matomaki
Kakeya conjecture
Lie algebras
Lie groups
Liouville function
Littlewood-Offord problem
Maksym Radziwill
Mobius function
multiple recurrence
Navier-Stokes equations
nilpotent groups
nilsequences
nonstandard analysis
parity problem
politics
polymath1
polymath8
Polymath15
polynomial method
polynomials
prime gaps
prime numbers
prime number theorem
pseudorandomness
random matrices
randomness
Ratner's theorem
regularity lemma
Ricci flow
Riemann zeta function
Roth's theorem
Schrodinger equation
sieve theory
spectral theorem
structure
Szemeredi's theorem
Tamar Ziegler
ultrafilters
universality
Van Vu
wave maps
Yitang Zhang
The Polymath Blog
- A sort of Polymath on a famous MathOverflow problem 9 June, 2019
- Ten Years of Polymath 3 February, 2019
- Updates and Pictures 19 October, 2018
- Polymath proposal: finding simpler unit distance graphs of chromatic number 5 10 April, 2018
- A new polymath proposal (related to the Riemann Hypothesis) over Tao’s blog 26 January, 2018
- Spontaneous Polymath 14 – A success! 26 January, 2018
- Polymath 13 – a success! 22 August, 2017
- Non-transitive Dice over Gowers’s Blog 15 May, 2017
- Rota’s Basis Conjecture: Polymath 12, post 3 5 May, 2017
- Rota’s Basis Conjecture: Polymath 12 6 March, 2017
9 comments
Comments feed for this article
14 September, 2009 at 11:26 am
Anonymous
Interesting! (There’s a typo in the third paragraph of the first section)
15 September, 2009 at 2:38 am
Manjil P. Saikia
Interesting, maybe the other talks like the one on Poincare Conjecture can also be made into such a format.
16 September, 2009 at 4:47 am
Thuris
Thank you for posting this. Although a non-mathematician, I found it very clear. (Which gives me hope to understand some of the other posts eventually!)
17 September, 2009 at 11:31 am
Hassan Mohy-ud-Din
Greetings Professor Terence Tao.
How be you? I am a PhD student at Johns Hopkins University. My request is that if you can kindly upload the videos of the talks/lectures along with the slides it will be very helpful and also make a tab of upcoming seminars/conferences/workshops. Thank You
Good Bye
17 September, 2009 at 8:09 pm
xi'an
Dear Terence, there is a typo in the references: de La Vallée Poussin’s book is “Recherches analytiques sur la théorie des nombres premiers”
Best regards,
Xi’an
30 September, 2009 at 8:27 am
zeynel
Yes, thank you for the slides. I enjoyed reading it as a non-mathematician. In your last slide you say that primes are *mostly* odd. Does this suggest that there may be even primes except 2?
30 September, 2009 at 1:47 pm
ARN
Zeynel , I think that was probably a joke. There can be no even primes > 2.
30 September, 2009 at 5:10 pm
links for 2009-09-30 « Blarney Fellow
[…] Structure and randomness in the prime numbers (IMO Festschrift submission) « What’s new (tags: random primes math complexity computation) […]
26 January, 2011 at 7:24 am
Tom Verhoeff
The grammar seems broken in the sentence containing “to not obey behave pseudorandomly”. You can delete “obey” to fix the grammar, but probably “not” should be deleted as well to fix the semantics. This is both on the slides and in the paper.
[Thanks! I think this was fixed in the published version of the text – T.]