All Questions
Tagged with big-list nt.number-theory
66 questions
239
votes
14
answers
76k
views
Have any long-suspected irrational numbers turned out to be rational?
The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
157
votes
7
answers
74k
views
Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...
131
votes
14
answers
30k
views
Why are modular forms interesting?
Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
117
votes
22
answers
39k
views
What's the "best" proof of quadratic reciprocity?
For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.
97
votes
19
answers
38k
views
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
87
votes
38
answers
26k
views
Where is number theory used in the rest of mathematics?
Where is number theory used in the rest of mathematics?
To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to ...
84
votes
31
answers
70k
views
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
75
votes
1
answer
12k
views
What if the Riemann Hypothesis were false?
There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?
67
votes
17
answers
12k
views
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...
- 7,127
57
votes
13
answers
20k
views
'Important' applications of p-adic numbers outside of algebra (and number theory).
Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily ...
54
votes
4
answers
3k
views
When has the Borel-Cantelli heuristic been wrong?
The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...
- 11.4k
52
votes
6
answers
5k
views
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?
One interesting example, and the impetus for this question, is work in number ...
48
votes
6
answers
5k
views
Are there examples of conjectures supported by heuristic arguments that have been finally disproved?
The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...
46
votes
2
answers
4k
views
What are the potential applications of perfectoid spaces to homotopy theory?
This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
45
votes
17
answers
49k
views
Good algebraic number theory books
I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic),...
43
votes
8
answers
21k
views
Approaches to Riemann hypothesis using methods outside number theory [closed]
Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The ...
42
votes
4
answers
6k
views
Famous vacuously true statements
I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states
Suppose that for each $...
40
votes
6
answers
4k
views
What motivations for automorphic forms?
Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this ...
36
votes
8
answers
32k
views
What are the connections between pi and prime numbers?
I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more ...
35
votes
9
answers
21k
views
Direct proof of irrationality?
There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is not of the form:
Suppose $a/b=\sqrt{2}$ ...
- 395
34
votes
21
answers
11k
views
Applications of finite continued fractions
I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...
33
votes
7
answers
3k
views
Examples of integer sequences coincidences
For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
33
votes
3
answers
3k
views
Arithmetic geometry examples
(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
30
votes
15
answers
17k
views
Useless math that became useful
I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...
30
votes
11
answers
5k
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 Euler'...
25
votes
3
answers
2k
views
Interpretations and models of permanent
The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix ...
- 13.9k
25
votes
2
answers
2k
views
Examples where the analogy between number theory and geometry fails
The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
23
votes
1
answer
3k
views
A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
19
votes
8
answers
3k
views
Open problems in continued fractions theory
I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
18
votes
7
answers
6k
views
Proofs of the Chevalley-Warning Theorem
A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...
- 1,890
18
votes
4
answers
621
views
What are immediate applications of the classification of connected reductive groups?
After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data.
That's a non-trivial theory! I'm hoping that now that I am done ...
- 557
17
votes
13
answers
6k
views
Probability in number theory
I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
17
votes
8
answers
4k
views
Algebraic number theory and applications to properties of the natural numbers.
Please allow me, for the purposes of this question(but only here), to exaggerate matters and state two polemic definitions. Please forget these definitions after answering this question, and pardon my ...
16
votes
2
answers
1k
views
Examples of problems where considering "discrete analogues" has provided insight or led to a solution of the original problem
The Kakeya conjecture posits that any Kakeya set in $\mathbb{R}^n$ has dimension $n$.
A discrete (finitized?) version of this problem is the Finite Field Kakeya conjecture, which was proved by Dvir ...
15
votes
7
answers
1k
views
Examples of proofs by making reduction to a finite set [closed]
This is a very abstract question, I hope this is appropriate.
Suppose $T$ is some claim over some infinite set $A$, for example, let $A$ be the set of all loopless planar graphs, and $T$ be the claim "...
14
votes
8
answers
2k
views
Applications of the idea of deformation in algebraic geometry and other areas?
The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
14
votes
2
answers
857
views
References for particular topics related to Langlands
I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...
13
votes
10
answers
1k
views
Properties of natural numbers such that there is a "very large largest" number with that property
I'm looking for properties (P) such that you would assume that there are infinitely many natural numbers with property (P) (for example because there are very large numbers with that property) but ...
13
votes
4
answers
1k
views
What results would follow from or imply "randomness" of the primes?
This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
12
votes
6
answers
3k
views
applications of Tate-Poitou duality
What are nice applications of Tate-Poitou duality?
12
votes
3
answers
849
views
Applications of idempotent ultrafilters
Recently Justin Moore has posted a solution to the amenability of Thompson's group F. A key(?) step exploits the existence of idempotent ultrafilters on $\mathbb N$ to construct an idempotent measure ...
12
votes
1
answer
565
views
On Bailey–Borwein–Plouffe formula for irrational numbers
A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...
11
votes
6
answers
3k
views
What are conjectures that are true for primes but then turned out to be false for some composite number?
Note: This is an update formulation since many people misunderstood the question before.
Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every ...
- 19k
11
votes
4
answers
1k
views
Compilation of strategies to show that some constant is irrational
I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...
- 521
10
votes
2
answers
2k
views
Consequences of Legendre's conjecture
I am looking for a list/reference which explores the consequences of Legendre's conjecture, which states that one can always find a prime number between $n^2$ and $(n+1)^2$.
- 215
10
votes
6
answers
733
views
Real number happens to be an integer
Sometimes you have a real number (with a rather complicated definition), and with some effort you can show that
1) this real number is, actually, an integer;
2) the distance of this real number to ...
9
votes
6
answers
5k
views
Examples of naturally occurring Quadratic forms or quadrics.
I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
9
votes
2
answers
791
views
Rational points techniques on curves not using their Jacobian
Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
- 2,224
9
votes
2
answers
1k
views
Adelic methods for classical modular forms
Many conjectures about properties of automorphic forms on $\mathrm{GL}(2)$ can be formulated in the basic language of classical modular forms (i.e. Hecke forms that are holomorphic on $\mathbb{H}$ or ...
- 8,374
9
votes
0
answers
3k
views
"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...