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

Filter by
Sorted by
Tagged with
17 votes
3 answers
2k views

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) = \...
Agno's user avatar
  • 4,179
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 ...
David S. Newman's user avatar
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 ...
David E Speyer's user avatar
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{...
Victor Miller's user avatar
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 ...
Frank Thorne's user avatar
  • 7,199
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{...
Daniel Soltész's user avatar
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 ...
Yufei Zhao's user avatar
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 $P(x)$, ...
Bogdan Grechuk's user avatar
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 ...
Stanley Yao Xiao's user avatar
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: $$\...
mmm's user avatar
  • 305
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 $\...
user42690's user avatar
  • 851
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 ...
István Kovács's user avatar
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 ...
pki's user avatar
  • 143
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 ...
Syang Chen's user avatar
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 ...
L. Milla's user avatar
  • 598
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 ...
David E Speyer's user avatar
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 $k$...
S. Li's user avatar
  • 619
13 votes
1 answer
3k views

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 ...
Thomas's user avatar
  • 131
13 votes
1 answer
1k views

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 ...
unramified's user avatar
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 $x$,$y$,...
Jernej's user avatar
  • 3,433
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)...
Julián Aguirre's user avatar
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 ...
Brian Rushton's user avatar
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+...
Benjamin Cornish's user avatar
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 $...
Stefan Kohl's user avatar
  • 19.5k
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 ...
amateur algebraist's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
Max Alekseyev's user avatar
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?
Luca's user avatar
  • 211
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 ...
user avatar
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 ...
Alexey Ustinov's user avatar
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 ...
Mark Lewko's user avatar
  • 11.7k
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?
darya's user avatar
  • 391
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 ...
Wanderer's user avatar
  • 5,113
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 ...
Pritam Majumder's user avatar
10 votes
2 answers
1k views

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}^{\...
user20174's user avatar
  • 439
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, ...
Andreas Rüdinger's user avatar
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}{(\...
Dr. Pi's user avatar
  • 2,939
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(\...
user avatar
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 ...
Stanley Yao Xiao's user avatar
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 ${\mathbb R}^2$). ...
Adam Sheffer's user avatar
  • 1,052
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} + \...
H A Helfgott's user avatar
  • 19.3k
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 $...
Tito Piezas III's user avatar
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^...
Tito Piezas III's user avatar
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 ...
Tito Piezas III's user avatar
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{\...
joro's user avatar
  • 24.2k
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 ...
Marty's user avatar
  • 13.1k
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 $\: \...
user avatar
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 ...
Bogdan Grechuk's user avatar
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 ...
Clark Kimberling's user avatar

1
3 4
5
6 7
46