# 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
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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?
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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 ...
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### 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 ...
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### “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, ...
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### 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: ...
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### 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 ...
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### “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 ...
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### 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 $\mathbb Q$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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-...
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### 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 ...
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### Any work on the Adams-Watters triangle?

Does anyone know whether any arithmetical or asymptotic results have been obtained about the Adams-Watters triangle?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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,...
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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, ...
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### What is the Beilinson regulator?

Trying to understand answer to this question. What is the (Beilinson) higher regulator of a number field?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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) ?
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### 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 <...
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### 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 ...
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### Is 8 the largest cube in fibonacci sequence?

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