**1**

vote

**2**answers

170 views

### Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?

**18**

votes

**2**answers

3k views

### Location of Archimedes' grave in Syracuse (math/archaelogy trivia)

This is really a question for our archaelogist friends, but I could not find an "archaelogy overflow" and some mathematicians might find it interesting.
In a few weeks I am giving a talk in which I ...

**11**

votes

**1**answer

565 views

### Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...

**19**

votes

**1**answer

1k views

### history of quaternion algebras

Who is responsible for the generalization of Hamilton's quaternions to other types of quaternion algebras, and when did this occur? In particular, Hamilton's quaternions are the 4-dimensional algebra ...

**17**

votes

**1**answer

409 views

### When did “Betti cohomology” come to be used the way it is today? (and how is it used)

This is sort of a mixture of a math and history question.
First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I ...

**11**

votes

**2**answers

536 views

### First formulation of the Dedekind and Hasse-Weil conjectures

I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
...

**31**

votes

**10**answers

6k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**52**

votes

**3**answers

4k views

### What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as ...

**63**

votes

**28**answers

9k views

### What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly ...

**4**

votes

**1**answer

187 views

### Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:
Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially?
...

**6**

votes

**1**answer

125 views

### How to divide a square into three similar rectangles

Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles?
With a bit of algebra it can ...

**9**

votes

**1**answer

352 views

### Examples of abstractions that did *not* turn out to be useful [closed]

I’ve read (but cannot find any reference now) that new abstract mathematical concepts like set theory and – not too long ago – category theory were in their time often considered too abstract to be ...

**68**

votes

**26**answers

10k views

### What are some famous rejections of correct mathematics?

Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner
Theorem: There are only a finite
number of imaginary quadratic fields
that have unique factorization. ...

**8**

votes

**2**answers

1k views

### How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematics

Does anyone know how news of Gödel's incompleteness theorem spread? Did it do so little by little, or was it shouted in dramatic headlines throughout mathematical literature? If anyone can point me to ...

**16**

votes

**2**answers

372 views

### Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...

**11**

votes

**1**answer

404 views

### Two Vinogradovs? Is one the son of the other? [closed]

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming ...

**20**

votes

**1**answer

653 views

### What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here.
After Ramanujan and ...

**16**

votes

**1**answer

1k views

### William Rowan Hamilton and Algebra as Time

This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the ...

**22**

votes

**1**answer

676 views

### How and why did mathematicians develop spin-manifolds in differential geometry?

First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...

**19**

votes

**19**answers

24k views

### Good books on problem solving / math olympiad

Hello,
I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would ...

**139**

votes

**136**answers

30k views

### Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...

**162**

votes

**36**answers

43k views

### Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

**11**

votes

**1**answer

891 views

### Taniyama's original conjecture

I've just read on Wikipedia that the original Taniyama conjecture about L-functions of elliptic curves over an arbitrary number field was still unproven.
This made me want to know more about this ...

**5**

votes

**2**answers

444 views

### Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus

The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the ...

**35**

votes

**4**answers

2k views

### Hilbert's (cancelled) 24th problem

Hilbert's 23 problems, ten of which were presented at the 1900 ICM in Paris, are too famous for any mathematician to not know. If one reads the descriptions of the problems in Hilbert's paper, one ...

**242**

votes

**72**answers

90k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**4**

votes

**1**answer

635 views

### Grothendieck's letter to Serre on the Standard Conjectures

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?

**90**

votes

**94**answers

12k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**2**

votes

**0**answers

115 views

### When did mathematicians begin to use the letter x to denote unknown values? [duplicate]

When did mathematicians begin to use the letter x to denote unknown values ?
Gérard Lang

**11**

votes

**1**answer

289 views

### 'Updated' book in the same spirit as Dieudonné's Panorama des mathématiques pures

Today a colleague of mine asked me if I knew of any "more modern version" of J. Dieudonné's Panorama des mathématiques pures. Le choix bourbachique.
The very first thing that instantly came to my ...

**6**

votes

**1**answer

135 views

### History of spectral methods to the study of real analytic $GL_2$-Eisenstein series

I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...

**21**

votes

**2**answers

1k views

### What might extraterrestrial mathematics look like? [closed]

In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely ...

**70**

votes

**21**answers

12k views

### Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...

**11**

votes

**2**answers

505 views

### Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs.
Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...

**57**

votes

**22**answers

8k views

### Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of
Major theorems in mathematics whose proofs ...

**60**

votes

**29**answers

6k views

### The half-life of a theorem, or Arnold's principle at work

Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that ...

**16**

votes

**4**answers

612 views

### History of powers beyond squares and cubes

The ancient Babylonians understood squares:
Plimpton 322
The ancient Athenians understood cubes, if we can take
doubling the cube, i.e., the Delian problem, as evidence.
My ...

**18**

votes

**2**answers

520 views

### History of set-class distinction

I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class:
Who was the first mathematician to make this ...

**64**

votes

**66**answers

10k views

### Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...

**1**

vote

**0**answers

208 views

### Why do some people adamantly insist on 'toposes' instead of 'topoi'? [closed]

I've heard that several category and topos theorists, first and foremost Johnstone (see the comments to this question) adamantly insist on 'toposes' as the plural of 'topos'. I was wondering whether ...

**19**

votes

**9**answers

2k views

### Do there exist modern expositions of Klein's Icosahedron?

Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?

**6**

votes

**1**answer

730 views

### Windows into new mathematical worlds [closed]

Yitang Zhang's Annals of Mathematics primes-gap result
opened a new window, which
Polymath's reduction from $70\times 10^6$ to $246$ attests.
Perhaps
Harald Helfgott's
celebrated proof of the odd ...

**56**

votes

**8**answers

6k views

### Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending ...

**11**

votes

**2**answers

749 views

### Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?

**8**

votes

**0**answers

210 views

### On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and ...

**44**

votes

**6**answers

6k views

### Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?

According to Steven Krantz's Mathematical Apocrypha (pg. 186):
As was custom, Weil often attended tea
at [Princeton] University . Graduate
student Steven Weintrab one day went
about the room ...

**1**

vote

**0**answers

124 views

### Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...

**1**

vote

**0**answers

657 views

### Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...

**10**

votes

**4**answers

1k views

### Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...

**54**

votes

**27**answers

8k views

### Writing papers in pre-LaTeX era?

I wonder how people wrote papers in the pre-LaTeX era? I mean, when typewriters and simple computers were (60th-70th?). Did they indeed put formulas by hand in the already printed articles?