# Tagged Questions

**10**

votes

**3**answers

2k views

### Why are smooth numbers called “smooth”?

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.)
Does anyone know what is the ...

**22**

votes

**2**answers

1k views

### Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective?
Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...

**6**

votes

**2**answers

848 views

### Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...

**0**

votes

**3**answers

543 views

### Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...

**7**

votes

**1**answer

257 views

### History question: Roth's theorem on approximating algebraic numbers…before Roth

Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality
...

**17**

votes

**1**answer

466 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 ...

**10**

votes

**2**answers

639 views

### Le Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides

Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in Roger Godement's Analyse mathématique, IV, p.313 (perhaps the only treatise on ...

**23**

votes

**0**answers

715 views

### Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...

**21**

votes

**2**answers

957 views

### The Erdős-Turán conjecture or the Erdős' conjecture?

This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ ...

**6**

votes

**1**answer

219 views

### Did Smith correctly state the mass formula?

Did Smith correctly state the mass formula?
H.J.S. "normal form" Smith was the first, in 1867, to state the mass formula for integral quadratic forms in a genus of 4 or more variables. This was ...

**27**

votes

**3**answers

2k views

### What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math.
The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...

**11**

votes

**1**answer

937 views

### At what point does number theory stop playing with finite rings?

Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for ...

**10**

votes

**1**answer

971 views

### Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below.
In ...

**5**

votes

**3**answers

1k views

### First known proof of $\sqrt 2$ is irrational with prime factorization?

Do any of you happen to know the history of the standard prime factorization proof of $\sqrt 2$ is irrational? I know this theorem was known to Aristotle, and that the Fundamental Theorem of ...

**29**

votes

**2**answers

1k views

### Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

(This question was posted on math.stackexchange a week ago at http://math.stackexchange.com/questions/187315/definitive-source-about-dirichlet-finally-proving-the-unit-theorem-in-the-sistinbut and ...

**12**

votes

**2**answers

666 views

### Did Hermite really prove “Hermite's Theorem” on number field discriminants?

Hermite's theorem, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant.
The usual proof (see Neukirch's Algebraic Number Theory ...

**3**

votes

**2**answers

469 views

### Who discovered the asymptotic formula for the number of partitions of n into distinct parts?

Who was the first to develop the asymptotic formulae for the distinct parts version of $p(n)?$

**5**

votes

**1**answer

255 views

### Who proved that the plane partition generating function is valid?

I know Major Macmahon conjectured the formula $$ \prod_{m=1}^\infty \frac{1}{(1-q^m)^m}=1 + \sum_{n=1}^\infty PL(n)q^n$$
but who was the first to prove it?

**14**

votes

**5**answers

956 views

### What is the “ray” in ray class group?

I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a ...

**3**

votes

**2**answers

987 views

### Heuristics for the Hodge Conjecture

W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.
I am ...

**6**

votes

**3**answers

972 views

### Sum of the sum-of-divisors function

I was looking at the abstract of a paper [1] which claims that [2] and [3] prove
$$
\sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x).
$$
But I cannot find the above—or indeed, ...

**7**

votes

**3**answers

2k views

### Riemann zeta at even integers

I am talking about this in a course I am teaching, and hence am wondering: what are the various derivations of the values of Riemann zeta function at even integers? There are two incredibly cool ...

**11**

votes

**5**answers

2k views

### sums of rational squares

It is a well-known fact that if an integer is a sum of two rational squares then it is a sum of two integer squares. For example, Cohen vol. 1 page 314 prop. 5.4.9. Cohen gives a short proof that ...

**31**

votes

**1**answer

1k views

### On a remark of Tait on FLT for the exponent 3

This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...

**7**

votes

**1**answer

382 views

### Two implicit references in Serre's *Groupes de Galois : le cas abélien*

In his exposé at the Galois bicentenary conference, Serre makes two references which are not quite explicit.
The first reference occurs (at 22:30 in the video) when he is talking about Dedekind's ...

**9**

votes

**3**answers

956 views

### The first complete proof of the Kronecker-Weber theorem

While the Kronecker-Weber theorem —that every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field— is always attributed to, well, Leopold Kronecker and Heinrich Martin Weber, ...

**6**

votes

**1**answer

2k views

### How did Birch and Swinnerton Dyer arrive at their conjecture?

I suspect that they knew that the $L-$function is defined only for $Re(s) \gt 3/2$. Did they attempt to evaluate the $L-$function at $s=1$ by plugging $s=1$ in the infinite product $\prod_p ...

**5**

votes

**1**answer

486 views

### Kummer and Fermat's Equation

In Report on the Theory of Numbers, H.J.S. Smith writes:
"The impossibility of solving [Fermat's] equation has been demonstrated by M. Kummer, first, for all values of $\lambda$ not included among ...

**5**

votes

**5**answers

2k views

### Exponential sums for beginner.

What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...

**11**

votes

**4**answers

8k views

### The Ramanujan Problems.

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here.
In the Wikipedia page on Ramanujan, there is ...

**19**

votes

**1**answer

1k views

### Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets):
a) compositions with parts from {1,2}
(e.g., 2+2 = 2+1+1 = ...

**23**

votes

**4**answers

2k views

### Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened.
The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...

**11**

votes

**2**answers

699 views

### Historical Articles about zeta functions of curves

Are there any historical articles about the origins of zeta functions of curves over global fields (undoubtedly starting with $\mathbb{Q}$)? In particular who (and when did this happen) first have ...

**14**

votes

**6**answers

2k views

### A non-technical account of the Birch—Swinnerton-Dyer Conjecture

I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was ...

**15**

votes

**1**answer

1k views

### Primes represented by two-variable quadratic polynomials

I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its ...

**13**

votes

**9**answers

2k views

### New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are:
First example is classical... which is ...

**2**

votes

**1**answer

975 views

### $230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan

Just browsing some old stuff in my office for other thing I found the following:
$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan.
Find three numbers such that their sum, the sum ...

**2**

votes

**1**answer

421 views

### Whence the k-tuple conjecture?

What is the source of the $k$-tuple conjecture, that every integer tuple $(k_1,\ldots,k_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst ...

**2**

votes

**2**answers

1k views

### What is the shortest proof of the existence of a prime between $p$ and $p^2$ ? other examples? [closed]

1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)?
(One can say that we can have it as a collorary of ...

**12**

votes

**2**answers

3k views

### Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?

Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?

**15**

votes

**2**answers

1k views

### History of Irrationality results

The Greeks knew that numbers of the form $\sqrt{n}$ for nonsquare
integers $n$ are not rational. Much later, Lambert (1768) proved that
the values of $e^x$ and $\tan x$ are irrational for nonzero ...

**30**

votes

**2**answers

2k views

### First correct proof of FLT for exponent 3?

It is well known that Euler gave the first proof of FLT ($x^n + y^n = z^n$ has
no nontrivial integral solutions for $n > 2$) for exponent $n=3$, but that his proof
had gaps (which are not as easily ...

**8**

votes

**3**answers

667 views

### English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...

**44**

votes

**3**answers

5k views

### who fixed the topology on ideles?

I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not ...

**17**

votes

**6**answers

1k views

### A gamma function identity

In some of my previous work on mean values of Dirichlet L-functions, I came upon the following identity for the Gamma function:
\begin{equation}
\frac{\Gamma(a) \Gamma(1-a-b)}{\Gamma(1-b)}
+ ...

**34**

votes

**3**answers

2k views

### What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...

**7**

votes

**7**answers

2k views

### Meaning of Kronecker's comment to Lindemann

At the Mactutor history page, it is said that Kronecker remarked to Lindemann:
"What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational ...

**2**

votes

**2**answers

496 views

### Fermat for polynomials, as used in the AKS (Agrawal-Kayal-Saxena) algorithm

The basis for the deterministic polynomial-time algorithm for primality of Agrawal, Kayal and Saxena is (the degree one version of) the following generalization of Fermat's theorem.
Theorem
...

**11**

votes

**5**answers

1k views

### Christening Fermat's Little Theorem

I am writing an article on Fermat's work in number theory and feel uncomfortable everytime I have to write "Fermat's Little Theorem": it's clumsy and belittles the fundamental character of Fermat's ...

**31**

votes

**3**answers

4k views

### How do we know that Fermat wrote his famous note in 1637?

It is widely stated that Fermat wrote his famous note on sums of powers ("Fermat's last theorem") in, or around, 1637. How do we know the date, if the note was only discovered after his death, in ...