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
2,270
questions
17
votes
3
answers
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Does this infinite sum provide a new analytic continuation for $\zeta(s)$?
It is well known that the infinite sum:
$$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$
only converges for $\Re(s)>1$.
The Dirichlet 'alternating' sum:
$$\displaystyle \zeta(s) = \...
16
votes
0
answers
452
views
A Product Related to Unrestricted Partitions
Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
16
votes
4
answers
2k
views
Arithmetic progressions without small primes
The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...
16
votes
2
answers
1k
views
Simple proofs for the existence of elliptic curves having a given number of points
Yesterday, after he gave a nice talk, Dick Gross and I were chatting and he brought up the following annoying problem: suppose that for $p$ a prime that $H_p$ is the "Hasse interval" $[p+1- 2 \sqrt{...
16
votes
4
answers
9k
views
Exact formulas for the partition function?
I am curious, what kind of exact formulas exist for the partition function $p(n)$?
I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely ...
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
16
votes
1
answer
2k
views
Simple recurrence that fails to be integer for the first time at the 44th term
The sequence defined by $a_0=a_1 =1$ and
$$
a_n = \frac{1}{n-1}\sum_{i=0}^{n-1}a_i^2, \quad n > 1
$$
fails to be integer for the first time at $a_{44}$. Why??
You can verify the statement by ...
16
votes
2
answers
1k
views
Representing $x^3-2$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
15
votes
3
answers
5k
views
Numbers with known irrationality measures?
For a given real number $x$, let $R_x$ be the set of real numbers $r$ such that the inequality
$$\displaystyle \left| x - \frac{p}{q} \right| < \frac{1}{q^r}$$
has at most finitely many solutions ...
15
votes
3
answers
974
views
Unit fraction, equally spaced denominators not integer
I've been looking at unit fractions, and found a paper by Erdős "Some properties of partial sums of the harmonic series" that proves a few things, and gives a reference for the following theorem:
$$\...
15
votes
1
answer
929
views
When is the image of a 2-dim l-adic representation associated to a modular form open
I know the following theorems by Serre:
1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open.
2, The 2-dim l-adic representation associated the weight-12 cusp form $\...
14
votes
1
answer
1k
views
Consecutive non squarefree integers
Question: Is there a function $f(n) \rightarrow \infty$, such that infinitely often the interval $[n,n+\frac{f(n) \log(n)}{\log{\log(n)}}]$ does not contain a squarefree integer?
Additional ...
14
votes
2
answers
2k
views
Number fields with same zeta function?
Given a zeta function $\zeta_K$ of some number field $K$ how much information will this give us about $K$? Specifically, if two number fields have the same zeta function, what shared properties are ...
14
votes
4
answers
2k
views
Fourier decay rate of Cantor measures
For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
14
votes
2
answers
1k
views
Complex Multiplication and algebraic integers
Let $q=e^{2\pi i\tau}$ and
$$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$
and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
14
votes
4
answers
3k
views
Does Weyl's Inequality prove equidistribution?
Let $f(n) = \theta n^d + a_{d-1} n^{d-1} + \cdots a_1 n + a_0$ be a polynomial with real coefficients, and $\theta$ irrational. Let $S_N = \sum_{n=1}^N e^{2 \pi i f(n)}$. Weyl's Equidistribution ...
13
votes
1
answer
1k
views
Classify all the fields with abelian absolute Galois group
I'm wondering if anyone has classified all the fields $K$ such that $Gal(\bar{K}/K)$ is abelian?
The only examples I'm aware of are: finite fields, the real numbers $\mathbb{R}$ and $k((T))$ where $k$...
13
votes
1
answer
3k
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p-adic Hodge theory for varieties defined over \C _p ?
I have a question on p-adic Hodge theory:
When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a ...
13
votes
1
answer
1k
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Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
13
votes
6
answers
10k
views
Frobenius number for three numbers
Given integers $a,b,c$ such that $\gcd(a,b,c) = 1$, it is well known that there exists only a finite set of numbers $n$ such that $n$ is not expressible as $ax+by+cz$ for non negative integers $x$,$y$,...
13
votes
4
answers
4k
views
When is the sum of two quadratic residues modulo a prime again a quadratic residue?
Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is
$$
\frac{p-6+\text{mod}(p,4)...
12
votes
2
answers
3k
views
Eigenvalues of nonnegative integer matrices
Edit
I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post:
What are the possible eigenvalues of nonnegative integer matrices?
Any answer to ...
12
votes
3
answers
8k
views
Is there an algorithm for writing a number as a sum of three squares?
By Gauss's Theorem, every positive integer $n$ is a sum of three triangular numbers;
these are numbers of the form $\frac{m(m+1)}2$. Clearly
$$ n = \frac{m_1^2+m_1}2 + \frac{m_2^2+m_2}2 + \frac{m_3^2+...
12
votes
0
answers
537
views
Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$...
11
votes
4
answers
12k
views
How to find all integer points on an elliptic curve?
How can I determine the integer points of a given elliptic curve if I know its rank and its torsion group?
I read same basic books on elliptic curves but as a non-professional I didn't understand ...
11
votes
2
answers
1k
views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
11
votes
1
answer
2k
views
Integer values of a rational function
Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the ...
11
votes
2
answers
2k
views
$3^n - 2^m = \pm 41$ is not possible. How to prove it?
$3^n - 2^m = \pm 41$ is not possible for integers $n$ and $m$. How to prove it?
11
votes
2
answers
1k
views
Does this product have analytic continuation?
The product
$$
F(s)=\prod_{p}\frac1{(1-p^{-s})^p},
$$
converges for $\mathrm{Re}(s)>2$, when $p$ runs over all primes. Does it admit analytic continuation beyond the line $\mathrm{Re}(s)=2$? Any ...
11
votes
1
answer
673
views
2-adic Logarithm and Resistance of n-dimensional Cube
Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=0}^{n-1}\frac1{{n-1\choose k}}.$$
The ...
11
votes
1
answer
2k
views
What is the best known upper bound for the number of twin primes?
A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2\, C\, n/ \ln^2(n)$ where $C$ is the so-called twin prime ...
11
votes
2
answers
1k
views
What is prime power of this equation of p?
Let $p$ be a prime number, I think when $p^2+p+1=q^a$, where $q$ is a prime number, then $a=1$. But I can't prove it. Is it true?
10
votes
2
answers
3k
views
adding an n-th root to Q_p
What can be said about extensions à la $\mathbb{Q}_p(\sqrt[n]{a})/\mathbb{Q}_p$? Ramification behaviour, valuation ring, ...?
I find it hard to say anything general - for example, as a function of ...
10
votes
7
answers
2k
views
Getting a bound on the coefficients of the factor polynomial
Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible ...
10
votes
2
answers
1k
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Reference request: Oldest number theory books with (unsolved) exercises?
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the ...
10
votes
1
answer
2k
views
Any progress on the Firoozbakht Conjecture? [closed]
Let $p_n$ be the n-th prime. The Firoozbakht Conjecture is a lesser known conjecture in the theory of primes but it has important consequences. It states that
$$
p_n^{\frac{1}{n}} > p_{n+1}^{\...
9
votes
3
answers
1k
views
Have new conjectures generated by the Ramanujan machine been proven?
Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
9
votes
1
answer
442
views
Error term in Davenport's sum $\sum_{n \leq x } \mu(n) \exp(2 \pi i \alpha n ) $
Reference request:
Davenport proved that for every fixed $N>1 $ one has $$ \sup_{\alpha \in \mathbb R } \left | \sum_{1\leq n \leq x } \mu(n) \exp(2 \pi i \alpha n )\right | = O_N\left( \frac{x}{(\...
8
votes
2
answers
616
views
The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$
I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
8
votes
2
answers
1k
views
On the class number
If $K = \mathbb{Q}(\alpha)$ is a number field, where $\alpha$ is algebraic, and $\mathcal{O}_K$ the ring of integers in $K$, then the set of fractional ideals over $\mathcal{O}_K$ forms a group and if ...
8
votes
1
answer
2k
views
Lattice points on the boundary of an ellipse
How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...
8
votes
2
answers
739
views
$L_2$ bounds for tails of $\zeta(s)$ on a vertical line
Let $0<\sigma\leq 1$. Let $T$ be large. How can we give good explicit $L^2$ bounds on the tails of $\zeta(\sigma+it)$? That is, we want to bound the quantity $$\int_{\sigma-i\infty}^{\sigma-iT} + \...
7
votes
2
answers
411
views
A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics
I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
7
votes
3
answers
757
views
Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins
(Update):
Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...
7
votes
1
answer
376
views
A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
7
votes
2
answers
854
views
Unexpectedly prime rich cubic polynomial
We got a cubic polynomial which is unexpectedly prime rich.
Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and
$\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.
Let $F(n)=\frac{\...
7
votes
1
answer
437
views
Algorithm for comparing $x + y \cdot \log(z)$ for $x,y,z$ rational?
I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures ...
7
votes
1
answer
766
views
Are there effective small intervals in which primes are dense?
As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: \...
7
votes
1
answer
419
views
$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
7
votes
2
answers
426
views
Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...