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
2,246
questions
26
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2
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Partitions to different parts not exceeding $n$
Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or ...
26
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6
answers
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Proofs of Jacobi's four-square theorem
What are the nicest proofs of Jacobi’s four-square theorem you know? How much can they be streamlined? How are they related to each other?
I know of essentially three aproaches.
Modular forms, as in,...
26
votes
7
answers
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When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of ...
26
votes
4
answers
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Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
26
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6
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Where can I find a comprehensive list of equations for small genus modular curves?
Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly interested in genus ...
26
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3
answers
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The diophantine eq. $x^4 +y^4 +1=z^2$
This question is an exact duplicate of the question
Does the equation $x^4+y^4+1=z^2$ have a non-trivial solution?
posted by Tito Piezas III on math.stackexchange.com.
The background of ...
26
votes
1
answer
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What is the status on this conjecture on arithmetic progressions of primes?
The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...
25
votes
2
answers
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The prime numbers modulo $k$, are not periodic
Consider the sequence of prime numbers: $2,3,5,7, \cdots$. Now reduce this sequence modulo $k$ for some integer $k > 2$. Show the resulting sequence is not periodic. :
EDIT: As noted in the ...
25
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2
answers
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Question on a generalisation of a theorem by Euler
We call an integer $k\geq 1$ good if for all $q\in\mathbb{Q}$ there are $a_1,\ldots, a_k\in \mathbb{Q}$ such that $$q = \prod_{i=1}^k a_i \cdot\big(\sum_{i=1}^k a_i\big).$$
Euler showed that $k=3$ is ...
25
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5
answers
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Monic polynomial with integer coefficients with roots on unit circle, not roots of unity?
There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example.
Now, for a monic polynomial of degree $n$, ...
25
votes
1
answer
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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 ...
24
votes
5
answers
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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 $\mathbb Q$, ...
24
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2
answers
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Do these rational sequences always reach an integer?
This post comes from the suggestion of Joel Moreira in a comment on An alternative to continued fraction and applications (itself inspired by the Numberphile video 2.920050977316 and Fridman, ...
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4
answers
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Does this sequence always give an integer?
It is known that the $k$-Somos sequences always give integers for $2\le k\le 7$.
For example, the $6$-Somos sequence is defined as the following :
$$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot ...
24
votes
3
answers
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Crux of Dwork's proof of rationality of the zeta function?
As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
24
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2
answers
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Why it is difficult to define cohomology groups in Arakelov theory?
I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says:
If $D$ is a divisor on $X$, we would like to define a ...
23
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3
answers
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How many different numbers can be obtained as product of first $n$ natural numbers?
Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot 2^{...
23
votes
1
answer
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The Dedekind eta function in physics
This interesting little fellow (a nice introduction is the video "Mock Modular Forms are Everywhere" by Cheng and Felder) popped up in some operator algebra (Witt / Virasoro Lie algebra) I ...
23
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3
answers
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Relation between Hecke Operator and Hecke Algebra
In the study of number theory (and in other branches of mathematics) presence of Hecke Algebra and Hecke Operator is very prominent.
One of the many ways to define the Hecke Operator $T(p)$ is in ...
23
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4
answers
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Does there exist a non-square number which is the quadratic residue of every prime?
I want to know whether there exist a non-square number $n$ which is the quadratic residue of every prime.
I know it is very elementary, and I think those kind of number are not exist, but I don't know
...
23
votes
1
answer
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Ramanujan's pi formulas with a twist
Given the binomial function $\binom{n}{k}$.
1. Define the following sequences,
$$\begin{aligned}
u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\
u_2(k) &...
23
votes
0
answers
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Is A276175 integer-only?
The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
22
votes
2
answers
2k
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unboundedness of number of integral points on elliptic curves?
If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...
22
votes
4
answers
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Special values of $p$-adic $L$-functions.
This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.
My understanding is that nowadays there are conjectures which essentially ...
22
votes
3
answers
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What is the relationship between modular forms and Maass forms?
Modular forms are defined here:
http://en.wikipedia.org/wiki/Modular_form#General_definitions
Maass forms are defined here:
http://en.wikipedia.org/wiki/Maass_wave_form
I wonder if modular forms can ...
22
votes
1
answer
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Little disks operad and $Gal (\bar {Q}/Q)$
My question is simple:
How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with ...
22
votes
4
answers
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Hasse principle for rational times square
Does a Hasse principle hold for the property of being a rational times a square ?
Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
22
votes
1
answer
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When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square?
This is a modification of an unanswered problem on the math StackExchange.
When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square?
If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n=3$, $k=...
22
votes
2
answers
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Is a real power series that maps rationals to rationals defined by a rational function?
Suppose that the function $p(x)$ is defined on an open subset $U$ of $\mathbb{R}$ by a power series with real coefficients. Suppose, further, that $p$ maps rationals to rationals. Must $p$ be defined ...
22
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3
answers
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A Collatz-like function that bifurcates on primes
This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...
22
votes
2
answers
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What is the matter with Hecke operators?
This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way.
However, what bother me is that ...
22
votes
4
answers
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Small quotients of smooth numbers
Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
21
votes
2
answers
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Geometric vs Arithmetic Frobenius
If an algebraic variety $X$ over a field characteristic p is given by equations $f_i(x_1,...,x_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients ...
21
votes
2
answers
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Implications of the disproof of the "climb-to-a-prime" conjecture
Now that James Davis has found a counter example, 13532385396179, to John Conway's climb-to-a-prime conjecture, I would be interested to learn whether this has any implications of interest in number ...
21
votes
1
answer
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What is the current status of the function fields Langlands conjectures?
My question, roughly speaking is, what happened to the function fields Langlands conjecture? I understand around 2000 (or slightly earlier perhaps), Lafforgue proved the function fields Langlands ...
21
votes
4
answers
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Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
20
votes
3
answers
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What is the probability that two numbers are relatively prime? [closed]
The basic question that I have is in the title, but let us make it more rigorous below.
Let $N=\{1, 2, ..., n\}$, and put the (normalized) counting measure, $\mu_n$, on $N\times N$.
Let $\mathcal{S}...
20
votes
1
answer
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Quantitative lower bounds related to Zhang's theorem on bounded gaps
Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{...
20
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2
answers
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Applications of number theory in dynamical systems
I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics.
...
20
votes
1
answer
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Is "almost-solvability" of Diophantine equations decidable?
Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...
20
votes
4
answers
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Splitting Pythagorean triples
Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would
be least surprised ...
20
votes
3
answers
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Size of set of integers with all sums of two distinct elements giving squares
Are there arbitrarily large sets $\mathcal S=\{a_1,\ldots,a_n\}$ of strictly positive integers such that all sums $a_i+a_j$ of two distinct elements in $\mathcal S$ are squares?
Considering subsets in ...
20
votes
2
answers
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Integral points on varieties
I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine ...
20
votes
3
answers
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Jensen Polynomials for the Riemann Zeta Function
In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in PNAS) the abstract includes
In the case of the Riemann zeta function, this proves the ...
20
votes
4
answers
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Establishing zeta(3) as a definite integral and its computation.
I am a 19 yr old student new to all these ideas. I made the transformation $X(z)=\sum_{n=1}^\infty z^n/n^2$. Therefore $X(1)=\pi^2/6$ as we all know (it is $\zeta(2)$). To calculate $X(1)$, I ...
19
votes
8
answers
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Can an algebraic number on the unit circle have a conjugate with absolute value different from 1?
I'm fearful about putting this forward, because it seems the answer should be elementary. Certainly, the Weak Approximation Theorem allows every system of simultaneous inequalities among archimedean ...
19
votes
2
answers
1k
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Floors of rationals to powers: Infinite number of primes?
Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).
Q. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\...
19
votes
3
answers
2k
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Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
19
votes
3
answers
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Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables ...
19
votes
2
answers
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New Geometric Methods in Number Theory and Automorphic Forms
The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
The branches of number theory most
directly related to ...