Kurt Heegner published his only, extremely influential paper, in 1952 when he was 59. However it took nearly 20 years for the mathematical community to realize what a gem it was.

And then there's the case of Serge Lang. Lang was an amazingly fast and accurate typist -- he could type at 90 words per minute. He developed an elaborate system using exacto knives, and glue. I ...

Persi Diaconis of Stanford had two careers -- the first as a violin prodigy studying at Julliard, and then as a world famous magician who performed for the crowned heads of Europe. In his early ...

The paper "Why You Cannot Even Hope to Use Gröbner Bases in Public-Key Cryptography? An Open Letter to a Scientist Who Failed and a Challenge to Those Who Have Not Yet Failed" by Boo Barkee , Julia ...

I'm surprised that nobody's mentioned almost anything by Emil Artin. His little monographs on Galois Theory and the Gamma Function are thrilling to read. They are so clear, and use the minimum ...

There's a slightly eccentric (but entertaining) book by Budden called "The Fascination of Groups" that has an extensive chapter about the application of groups to the British practice of "change ...

Barry Mazur "On Embeddings of Spheres", Bull. AMS v 65 (1959) only 5 1/2 pages. It introduced the method of infinite repetition in topology and allowed the proof the generalized Schoenflies ...

Another surprising connection is the Ax-Kochen theorem. Let $\mathcal{F}_{p,n,d}$ denote the set of homogeneous polynomials ("forms") in $n$ variables over the $p$-adics $\mathbb{Q}_p$ of ...

The idea (from the Selberg-Delange) method to doing this problem is the following steps: 1) Let $F(s) = \sum_{n\ge 1} \frac{1}{n^s d(n)} = \prod_{p} \left(1 + \sum_{k=1}^{\infty} \frac{1}{(k+1) p^{ks}...

The theory (and practice) of automatic groups is the most generally useful systematic way to deal with these things. There is a nice package written by Derek Holt and his associates called kbmag (...

There's an old easy proof of the fact that there are infinitely many primes $p$, $p \equiv 1 \bmod n$: Let $\Phi_n(X)$ be the $n$-th cyclotomic polynomial. Show that $\Phi_n(X)$ has a root in $\...

I recommend that you read the second chapter of this book https://www.springer.com/gp/book/9783211774168, entitled Algorithms in Invariant Theory. In particular it shows that you you can use Groebner ...

There are a few different approaches to this: 1) As Peter Shor mentioned you can use a $3^n$ point transform with Bluestein's algorithm. 2) Even though there are no $2^n$ roots of unity there are ...

I'll put in a plug for my original paper with Lagarias and Odlyzko, as well as a recent paper by Bach, Klyve and Sorenson: http://www.ams.org/journals/mcom/2009-78-268/S0025-5718-09-02249-2/home.html ...

There is also the paper by Broker, Lauter and Sutherland "Modular polynomials via isogeny volcanoes" http://arxiv.org/abs/1001.0402 which gives a fast (in practice) algorithm to calculate modular ...

As the previous answers have stated the map from $F_n \rightarrow n$ is essentially a logarithm. Since the binary representation of $F_n$ has about $c n$ bits (for the appropriate constant $c = \...

In answer to the question as to whether $\text{tr}(C^n)$ is dense in $(-2,2)$: Choose $z=\exp(2 \pi i \theta)$ where $\theta$ is irrational, and let $C$ be the diagonal matrix with $z$ and $\overline{...

This is actually fairly straightforward, and reduces to the fact that $ \sum_{p \text{prime}} \frac 1p = \log \log x + C + o(1)$ for some constant $C$. To see how to apply this to the original ...

To make things concrete, suppose that we're working in characteristic $\ne 2,3$ (we can do something similar in those cases, though it gets messier), and the equation of the curve is $E : y^2 = x^3 + ...

The theory of automatic groups may be a help here. There is a nice package written by Derek Holt and his associates called kbmag (available for download here: http://www.warwick.ac.uk/~mareg/download/...

You should look at Carl Pomerance's follow-up paper: Ruth-Aaron pairs revisited, http://www.math.dartmouth.edu/~carlp/PDF/paper130.pdf . In his first paper with Erdös they proved a result which ...

The paper by Ron Graham and Bruce Rothschild which gives a really short proof (involving a complicated triple induction) of van der Waerden's theorem: R.L. Graham and B.L. Rothschild, A short proof ...

Dave Dummit's paper "Solving Solvable Quintics" http://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079014-X/S0025-5718-1991-1079014-X.pdf constructs a sextic out of the coefficients of the ...

You should look at Andrew Granville's survey http://www.dms.umontreal.ca/~andrew/PDF/GoldbachFinal.pdf . Among other things he talks about the behavior of $r_2(N)$ (or a suitably weighted version of ...

You should look at the paper by Erdos and Odlyzko "On the density of odd integers of the form $(p − 1)/2^n$ and related questions" in Journal of number theory, vol. 11 (1979) pp 257-263. Among other ...

I don't believe that the problem is NP-complete, because you're working in a fixed dimension. Since most of us believe that the hardest case is when the angle between the two lines is very small, ...

For univariate polynomials you should look at "An Efficient Algorithm for the Complex Roots Problem" by Andy Neff and John Reif http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=...

If you're interested in families of polynomials that don't have Galois group $S_n$ you should look at Noam Elkies' web page: "Trinomials $ax^n+bx+c$ with interesting Galois groups" about ...

Fouvry showed that the relative density is positive of primes $p$ for which the largest prime factor of $p+a$ is $\ge p^{\alpha}$ for $\alpha \approx .6687$. Etienne Fouvry, Théorème de Brun-...