All Questions
22 questions
60
votes
1
answer
6k
views
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?
Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
- 7,127
32
votes
2
answers
3k
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 $A\subset \mathbb{N}$ and $$\...
- 11.4k
31
votes
5
answers
8k
views
Fermat's proof for $x^3-y^2=2$
Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original ...
- 3,866
29
votes
1
answer
3k
views
The Riemann zeros and the heat equation
The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_{...
- 11.1k
26
votes
2
answers
3k
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. ...
- 20.2k
23
votes
2
answers
2k
views
Dirichlet and the prime number theorem
I browsed Dirichlets Werke today and was kind of surprised by two remarks that he made on p. 354 (Über die Bestimmung ...) and p. 372 (Sur l'usage ...). In the second paper, he claims (my ...
- 32.6k
21
votes
3
answers
3k
views
Twin Prime Conjecture Reference
I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...
- 1,588
17
votes
5
answers
4k
views
Fermat numbers and the infinitude of primes
Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.
In ...
- 8,842
12
votes
1
answer
563
views
reference request: rational points on the unit sphere
I wonder what would be a good/early reference for the fact:
rational points on the unit sphere (centered at the origin) are dense.
Stereographic projection (from a rational point in the sphere) ...
- 10.7k
9
votes
2
answers
709
views
Egyptian number theory
Might there be a good historical reference on Egyptian number theory ($ \sim 2000$ B.C.)? The following online reference by a professor at the UCLA indicates that they were aware of the Pythagorean ...
- 3,871
9
votes
0
answers
462
views
Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?
Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...
- 15.6k
8
votes
3
answers
1k
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 ...
- 315
8
votes
1
answer
595
views
Why was the factor $\frac12$ introduced in the Riemann $\xi$ function?
The factor $\frac12$ in the Riemann $\xi$ function:
$$\xi(s)=\frac12 s(s-1)\,\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)$$
was introduced by Riemann, however appears to be redundant. Once he had arrived at:
...
- 4,169
7
votes
3
answers
611
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
- 8,842
7
votes
2
answers
615
views
Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
7
votes
1
answer
573
views
Euler's Triangular Number closure properties
Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:
If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.
R. F. Jordan in the J. of ...
- 233
6
votes
2
answers
713
views
Origin and variations of problem on $4xy-x-y$ being square
One of the forms in which the Diophantine equation in question can be found in the literature is this:
Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
- 8,842
6
votes
1
answer
636
views
The history and original paper of the Rosser–Iwaniec sieve
I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated ...
- 559
6
votes
0
answers
346
views
When did the main conjecture in Vinogradov's mean value theorem first appear in literature?
Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...
- 71
5
votes
0
answers
246
views
Video abstracts for mathematical papers
I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...
- 1,451
5
votes
0
answers
169
views
Where does the notation $\operatorname{Tr}(\cdot)\bmod \ell^\alpha$ implies isomorphism come from?
In J-P Serre's article on Faltings-Serre (Resume du Course 1984-1985) he states (without proof) that for two finite-dimensional $\ell$-adic Galois representations of $\operatorname{Gal}(\mathbb{Q})$, ...
- 2,429
2
votes
1
answer
334
views
Intuition behind the proof of key step in Minkowski's second inequality on successive minima
I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...
- 8,842