22 Original Math Papers You Can Read for Free

The sum 1 + 1/4 + 1/9 + 1/16 + … had been keeping the best mathematicians in Europe busy since 1650 when the 28-year-old Euler showed it equals π²/6. By today’s standards his method doesn’t count as a full proof, because he factors the sine function by its roots as if it were a polynomial of infinite degree and never explains why that works, but that nerve is exactly what makes the paper worth reading.

The scanned Latin original sits in the archive, and even without Latin you can follow the formulas.

This is the paper showing that evenly spaced runs of primes like 3, 5, 7 or 5, 11, 17, 23, 29 exist as long as you want them. Green and Tao don’t study the primes directly to get there, they place them inside a better-behaved set that contains the primes densely enough and apply a version of Szemerédi’s theorem to that set.

That intermediate set is the real trick of the proof, and the whole paper is free on arXiv.

In 1900, at the International Congress of Mathematicians in Paris, Hilbert lists the problems the new century will have to deal with. In the talk itself he covers only 10 of the 23, and the full set appears in the printed text. First on the list is the continuum hypothesis, eighth is the Riemann hypothesis, tenth is the search for a general method that decides whether an equation has integer solutions, and many of the papers on this page are in fact answers given to that talk later on.

Mary Winston Newson’s English translation from 1902 is free.

Turing writes this paper to answer Hilbert’s decision problem, the question of whether there is a mechanical method that decides if any mathematical statement is true or false. The answer is no, but on the way there he describes the abstract machine we now call a Turing machine and the universal machine that can imitate every other machine. Every computer, your phone included, is an instance of that universal machine.

The paper has a few mistakes, too, and Turing publishes a short correction the following year. The full text is available as a PDF.

The question of whether every problem whose solution is easy to check is also easy to solve, the P versus NP problem, takes its first shape in these eight pages. Cook shows that a single problem about logical formulas is as hard as every problem in that class, so anyone who solves that one problem fast has solved them all fast. Leonid Levin reaches the same result independently in the Soviet Union.

It is still one of the six unsolved Millennium Problems, and a scan of Cook’s paper survives in its two-column typesetting.

At 21, Shannon shows that the relay circuits in telephone exchanges can be written in Boolean algebra and simplified that way. Thinking of open and closed switches as 0 and 1 feels completely natural today, but before this thesis digital circuit design was done largely by trial and error.

The scanned thesis can be downloaded with its typewritten pages and hand-drawn circuit diagrams.

It opens with the question of whether machines can think, and instead of wrestling with definitions Turing turns the question into a game. An interrogator talks to a human and a machine through written messages, and if he can’t tell which one is the human, the machine is taken to have passed. Turing predicts that by the year 2000 an average interrogator will have no more than a 70 percent chance of guessing right after five minutes of conversation.

For me the most enjoyable part is the section where he lists the objections one could raise against a thinking machine and answers them one by one, and there is even one devoted to telepathy. A scan of the first printing in Mind is free.

Diffie and Hellman say a public-key encryption system is possible but don’t give a working encryption method. Rivest, Shamir and Adleman find the first working system, built on how hard it is to factor large numbers. Alice and Bob also make their first appearance in this paper, and cryptography has been explained through those two ever since.

The full text sits on Rivest’s page.

The question of whether there is a walk that crosses each of Königsberg’s seven bridges exactly once reaches Euler in a letter. Euler finds the question trivial in itself but finds it interesting that the answer has nothing to do with lengths and everything to do with what connects to what, and he solves it by giving the land masses capital letters and the bridges lowercase ones.

There isn’t a single graph drawn in this paper, which counts as the beginning of graph theory, and what Euler draws is the city itself. The Latin original is scanned in the archive.

The shortest paper on the list. At 21, Nash shows in two pages that every finite game has an equilibrium in which no player can gain by changing his own strategy while the others keep theirs fixed. The proof rests on Kakutani’s fixed point theorem and is nearly a single paragraph.

This is the idea that brought him the Nobel in economics in 1994, and the story around it is the backbone of A Beautiful Mind. Both of the two pages are free.

Maxwell ties electricity and magnetism into a single system of equations, then computes the speed of the waves that come out of those equations, and when he sees the result is almost exactly the measured speed of light he concludes that light is an electromagnetic wave too.

The Maxwell equations we write in four lines today, one of the 17 equations that changed the world, stand here as 20 separate equations, because vector notation doesn’t exist yet and Heaviside is still 20 years away from putting them in their present form. The 1865 printing is free to read.

In 1611 Kepler guesses that stacking spheres the way oranges are stacked at the grocer’s is the tightest way to pack them in three dimensions. Hales proves it 387 years later, but because the proof involves hundreds of pages of text and gigabytes of computer calculation, the referees can only say they are 99 percent sure even after years of checking. Hales then starts the Flyspeck project to translate the proof into a formal language a computer can check line by line, and finishes it in 2014.

The overview paper of the series gives the plan of the proof in its clearest form.