# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

10,877 questions

**26**

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### Are there two non-isomorphic number fields with the same degree, class number and discriminant?

If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?

**13**

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### Very strong multiplicity one for Hecke eigenforms

In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are ...

**31**

votes

**1**answer

3k views

### Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...

**4**

votes

**2**answers

563 views

### “half arithmetic progressions” in dense sets

Fix a positive real number d>0. Szemeredi's theorem implies that for every integer k, there exists an integer N(k,d) such that if A is a subset of the interval [1,N] with density greater than d >0, ...

**20**

votes

**1**answer

15k views

### How hard is it to compute the Euler totient function?

Are there any efficient algorithms for computing the Euler totient function? (It's easy if you can factor, but factoring is hard.)
Is it the case that computing this is as hard as factoring?
EDIT: ...

**44**

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**6**answers

5k views

### Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...

**5**

votes

**1**answer

460 views

### Finiteness of Obstruction to a Local-Global Principle

Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not ...

**48**

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**3**answers

3k views

### What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...

**12**

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**2**answers

591 views

### Number theoretic sequences and Hecke eigenvalues

What are some number theoretic sequences that you know of that occur as (or are closely related to) the sequence of Fourier coefficients of some sort of automorphic function/form or the sequence of ...

**4**

votes

**3**answers

1k views

### Modular forms reference

If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f.
I've seen this proven in ...

**11**

votes

**4**answers

623 views

### Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...

**12**

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**3**answers

1k views

### How should I approximate real numbers by algebraic ones?

Given a high precision real number, how should I go about guessing an algebraic integer that it's close to?
Of course, this is extremely poorly defined -- every real number is close to a rational ...

**6**

votes

**3**answers

404 views

### Prime numbers and strings of symbols

Suppose you have N symbols (e.g. "1","2",...,"N" or "a","b",...,"$") and a string of these symbols (say, the first trillion digits of pi). Then does there exist a prime number whose N-ary ...

**19**

votes

**1**answer

3k views

### Equivalent forms of the Grand Riemann Hypothesis

I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions.
In the case of the ordinary Riemann hypothesis, one gets a very good error term for the prime ...

**86**

votes

**10**answers

12k views

### “Understanding” $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...

**20**

votes

**5**answers

3k views

### Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$

The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals $\mathbb Q$, ...

**6**

votes

**2**answers

395 views

### Strong Bertrand postulate

Is it known that for every epsilon there is N_0 such that all intervals of the form [N, (1+\epsilon)*N], where N > N_0, contain ...

**3**

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**7**answers

2k views

### Bertrand's postulate [closed]

I believe there was an old conjecture that there's always a prime number between N and 2N.
What's the history and how is this ...

**11**

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**3**answers

2k views

### Is there a known bound in prime gaps?

Is there known to be an $x$ such that for all positive integers $N$ there exists some $n>N$ such that $p_{n+1}-p_n \leq x$, where $p_n$ is the $n$th prime? Or, in other words, is it known that ...

**1**

vote

**1**answer

903 views

### Neukirch's class field axiom and cohomology of units for unramified extension

This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the ...

**13**

votes

**3**answers

3k views

### What is Eisenstein series?

There are several related questions here, the latter being especially interesting. We know the classical Eisenstein series.
What are the Eisenstein series on a group G and why they are interesting?
...

**11**

votes

**4**answers

859 views

### Abel's equation for the dilog

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms)
plays a role in web geometry as it is one of the abelian relations of the
first example of exceptional web (Bol's 5-...

**14**

votes

**6**answers

1k views

### Solving polynomial equations when you know in which number field the solutions live

Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...

**3**

votes

**1**answer

385 views

### Any work on the Adams-Watters triangle?

Does anyone know whether any arithmetical or asymptotic results have been obtained about the Adams-Watters triangle?

**16**

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**3**answers

2k views

### 2-adic Coefficients of Modular Hecke Eigenforms

Suppose that $N$ is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level $\Gamma_0(N)$.
For such an eigenform $f$, the coefficients generate (an order in) the ring of ...

**10**

votes

**2**answers

875 views

### A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...

**11**

votes

**1**answer

685 views

### Covering the primes by 3-term APs ?

Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k.
My question is: can we actually partition the primes into 3-term APs only (or is there a ...

**5**

votes

**3**answers

562 views

### Solving “a, b, a+b have given divisors” problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...

**10**

votes

**3**answers

646 views

### A “round” lattice with low kissing number?

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...

**6**

votes

**1**answer

1k views

### Learning about Galois representations

My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.
Thus take pt = Spec F_q, ...

**7**

votes

**2**answers

2k views

### What is the Beilinson regulator?

Trying to understand answer to this question.
What is the (Beilinson) higher regulator of a number field?

**5**

votes

**2**answers

328 views

### Density of a subset of the reals

The rationals are clearly dense in the real number system, i.e. for every pair a < b of real numbers there exists a rational number p/q s.t. a < p/q < b. I conjecture the same to be true with ...

**35**

votes

**10**answers

5k views

### Why are functional equations important?

People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a ...

**35**

votes

**5**answers

7k views

### Definition and meaning of the conductor of an elliptic curve

I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an ...

**7**

votes

**3**answers

1k views

### p-power roots of unity in local fields

Let $K$ be a number field and suppose $K$ contains no $p$-power roots of unity. Let $\mathcal{P}$ be a prime of $K$ above the rational prime $p$. Can someone prove or disprove the assertion that the ...

**19**

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**3**answers

2k views

### Regulators of Number fields and Elliptic Curves

There is supposed to be a strong analogy between the arithmetic of number fields and the arithmetic of elliptic curves. One facet of this analogy is given by the class number formula for the leading ...

**26**

votes

**4**answers

2k views

### Why do zeta functions contain so much information?

Is there some intuitive explanation why Dedekind zeta functions contain so much information about their number field?
For example the residue at the pole $s=1$ relates several invariants of the ...

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vote

**2**answers

491 views

### Is an nth root of unity a square?

Suppose w^(2n)=1 (w is a complex number).
For which n (if any) \sqrt(w) \in Q(w) ?

**7**

votes

**2**answers

719 views

### How can I prove that a sequence of squares of graph norms is never cyclotomic?

The norm of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||G|| for the norm of G.
Now, fix some graph <...

**7**

votes

**1**answer

448 views

### Lifting bases for (Z/pZ)^n to Z^n

The following question came up in my research. I suspect that it has a slick answer,
but I can't seem to find it.
Fix an integer n>=2 and a prime p. Define X(n) to be the set of primitive
vectors ...

**24**

votes

**3**answers

8k views

### Is 8 the largest cube in fibonacci sequence?

Can you prove that 8 is the largest cube in fibonacci sequence?

**0**

votes

**7**answers

1k views

### What is the smallest integer whose primality status is not known?

Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current ...

**2**

votes

**7**answers

5k views

### What's so special about transcendental numbers? [closed]

It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them?
Are most numbers ...

**90**

votes

**22**answers

27k 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.

**25**

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### Global fields: What exactly is the analogy between number fields and function fields?

Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...

**7**

votes

**2**answers

4k views

### What does “supersingular” mean?

Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...

**7**

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**6**answers

2k views

### Primes are pseudorandom?

I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...

**26**

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**5**answers

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### Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s.
There's a very interesting article by ...

**7**

votes

**1**answer

562 views

### Ways to characterize supersingular primes?

I've read the definition, and it basically says p is a supersingular prime iff
the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.
And there's a ...

**11**

votes

**4**answers

1k views

### Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it,
...