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Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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When is $\varphi(n)$ one less than $\omega(n)$

When is $$\varphi(n) = \omega(n)+1$$ for $$1<n<1000$$ where $\varphi(n)$ represents numbers not exceeding $n$ coprime to $n$, and $\omega(n)$ represents numbers not exceeding $n$ that do not ...
Phoenix's user avatar
  • 19
0 votes
0 answers
41 views

Prove that for $n$ sufficiently large and $k \leq n$, there exists $m$ having exactly $k$ divisors $\le n$

Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$. here is ...
Saucitom's user avatar
1 vote
1 answer
78 views

Weierstrass Form of degree 4 equation

Take the equation $$y^2 = x^4 - 2x^3 - 2x - 1$$ I found that this is a genus 1 curve, because it is well known that for $y^2 = f(x)$ where $f$ is of even degree, the genus is $\frac{\deg{f} - 2}{2}$, ...
Ravikanth Athipatla's user avatar
9 votes
3 answers
294 views

$a^3 + b^3 + c^3 = 4abc$ has no positive integer solutions

Prove that the equation $a^3 + b^3 + c^3 = 4abc$ has no solutions in positive integers. Some attempts could be found at https://artofproblemsolving.com/community/q1h2213995p16779355 but none of them ...
DesmondMiles's user avatar
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1 vote
2 answers
135 views

What is the identity of this zeta function?

There are a Riemann zeta function, a Hurwitz zeta function, and many different types of zeta functions. However, I saw the zeta function below in a Japanese blog. $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{m=...
user1274233's user avatar
1 vote
0 answers
45 views

Prime-Independence of p-adic Continued Fractions: New Observation?

I've recently been exploring p-adic continued fractions and stumbled upon an intriguing pattern. It seems that the p-adic continued fraction representations of rational numbers are consistent across ...
P-Adic's user avatar
  • 11
1 vote
0 answers
43 views

Analogue of roots of unity in n-sphere

The $n$-th roots of unity $z_1,…,z_n$ in $\mathbb{C}\equiv \mathbb{R}^2$ for $n$ prime have an interesting property: for $0\leq p<n$ and $u$ a unit vector, the sum $$\sum_{k=1}^n \langle z_k,u\...
kaleidoscop's user avatar
6 votes
1 answer
68 views

Can we (almost always) walk from one Gaussian non-prime to another?

This is a plot of the Gaussian primes. They get sparser as you move further from $0$, so it looks like if you start on one of the white squares you could travel to any other white square (almost) ...
Zoe Allen's user avatar
  • 5,633
1 vote
1 answer
143 views

Given the primes, how many numbers are there?

I have a concrete question and a more abstract version of the same. There are lots of theorems that take information on the number of elements and translate it to information on the primes; I’m ...
Charles's user avatar
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-1 votes
1 answer
70 views

How to give this sum a bound?

Let $x,y\in\mathbb{Z},$ consider the sum below $$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$ is there anything I could do to give this sum a ...
Chang's user avatar
  • 329
0 votes
0 answers
38 views

Matrix algebras with involutions

This question relates to Example 8.5 in Milne's Introduction to Shimura Varieties. Let me set up some notation first. Let $k$ be a field of characteristic zero, and $B$ over $k$ an (not necessarily ...
Coherent Sheaf's user avatar
1 vote
0 answers
45 views

Finding all completely multiplicative arithmetic function such that m+n|f(m)+f(n)

My attempt: Since f is completely multiplicative we have $f(1)=1$. $2n|2f(n)$ for every n, so $f(n)=kn$ for some n. $n+1|f(n)+1$ for every n,so for p prime, $p+1|kp+1\rightarrow p+1|k-1$, so $k=l(p+1)+...
Dailin Li's user avatar
1 vote
0 answers
49 views

What condition on an abelian variety ensures that the associated formal group has integral coefficients?

I am new in the study of abelian variety or in general algebraic variety. I am looking for a sufficient condition that ensures the formal group associated with an abelian variety has integral ...
NT2024's user avatar
  • 11
2 votes
0 answers
65 views

Conjecture on Infinitely Many Consecutive Pairs of Early Primes

An early prime is one which is less than the arithmetic mean of the prime before and the prime after. Conjecture: There are infinitely many consecutive pairs of early primes MY attempt Well, the fact ...
Saucitom's user avatar
0 votes
0 answers
34 views

The solutions of $(2p^x)^{\varphi(2p^x)}+z^{\varphi(z)}=(2q^y)^{\varphi(2q^y)}$, being $\varphi(n)$ the Euler's totient.

The solutions of $(2p^x)^{\varphi(2p^x)}+z^{\varphi(z)}=(2q^y)^{\varphi(2q^y)}$, being $\varphi(n)$ the Euler's totient such that $p \neq q$ are primes, $v_2(z)=1$ and $\exists \ r_1,r_2 : r_1 \mid z \...
The Revolution's user avatar
3 votes
0 answers
84 views

Totally $p$-adic elements in cyclotomic extensions of $\mathbb{Q}$

Let $p$ be a prime number and fix $\bar{\mathbb{Q}}$ an algebraic closure of $\mathbb{Q}$. An element $\alpha \in \bar{\mathbb{Q}}$ is called totally $p$-adic if $p$ splits completely in $\mathbb{Q}(\...
cartesio's user avatar
  • 431
-3 votes
0 answers
28 views

What is your opinion about algebraic structure of primorials? [closed]

I did some exploration about primorial numbers in the past months. I published the findings as posts on my Linkedin page (Vladimir Lukovac). I find that primorial numbers have interesting ring ...
vlukovac's user avatar
1 vote
1 answer
57 views

How to check for solutions of non-linear systems over the integers modulo m?

We have the following system of 3 equations and 5 unknowns: $\left\{\begin{array}{lcl} u^2 + w^2 + x^2 + y^2 & = & 0\\ w^2 + x^2 + y^2 + z^2 & = & 1 \\ u^2 + x^2 + y^2 + z^2 & = &...
ERMTMG's user avatar
  • 11
-1 votes
0 answers
101 views

finding integer solutions to this polynomial problem [closed]

For what integer value(s) of x, do we get at least a rational number value for y in the polynomial equation below ?: $2y^3 - (6x+3)^2\cdot y^2 + (216x^4 + 432x^3 + 324x^2 + 57x - 12)\cdot y - (36x^3 + ...
A. J.'s user avatar
  • 31
1 vote
1 answer
40 views

A question about integrality in polynomial rings

Question: Let $k$ be any field and let $A:=k[x_1,\ldots,x_n ]$. Let $\bar{k}$ denote an algebraic closure of $k$, and let $B:=\bar{k}[x_1,\ldots,x_n]$. Show that the extension $B/A$ is integral. My ...
KaleBhodre's user avatar
  • 1,101
7 votes
1 answer
714 views

What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?

I learned in my intro to complex analysis class that the Riemann Zeta has trivial zeros at the negative even integers; this follows from the Riemann Zeta Reflection identity and the behavior of the $\...
Faraz Masroor's user avatar
-1 votes
0 answers
28 views

Integral representation of continued fractions [closed]

In $q$-series and allied areas, one try to express a continued fraction in terms of definite integrals. Ramanujan given integral representation of Rogers-Ramanujan continued fraction. In Ramanujan's ...
Sangama's user avatar
  • 21
2 votes
0 answers
44 views

Example of non supersingular Weil number

This question aims mainly at some misconception I may have regarding the formalism of Weil numbers. These are defined as some algebraic integers with a fixed complex modulus under all possible ...
Suzet's user avatar
  • 5,571
2 votes
1 answer
55 views

The "Euler Product formula" for general multiplicative functions

For the totient function $\phi$, we have the well known "Euler's product formula" (as named on Wikipedia) $$\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$$ This is easy to show ...
Instagram-creative_math_'s user avatar
0 votes
0 answers
23 views

Density or growth rate for Eisenstein integers by products and doing $2 x + k$

Consider these two sets of Eisenstein integers. SET 1 : constructed by these rules : a) any unit is in the set. b) if $x$ is in the set, then so is $2 x + 7$. c) if $x$ and $y$ are in the set then so ...
mick's user avatar
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1 vote
0 answers
47 views

Continued fraction of irrational number and periodicity

Theorem: Let $\xi$ be irrational. Its continued fraction $\xi = [a_0, a_1, \ldots]$ is periodic from a certain point, if and only if $\xi$ is an algebraic number of degree 2 (i.e., an irrational root ...
marek's user avatar
  • 15
0 votes
0 answers
15 views

Value of $k$ that gives the highest Restricted-Part Integer Partition Number for $n$

Let $p_k(n)$ be the number of possible partitions of an Integer $n$ into exactly $k$ parts. We know that for any given $n$, $p_k(n)$ gives a non-zero result for $0<k\leq n$, and that the size of ...
Lee Davis-Thalbourne's user avatar
-1 votes
1 answer
60 views

Solution of $3^{b+1} = 2^{n+2} - 5$ and $3^{b+1} = 2^{n+2} + 1$ in natural numbers [closed]

In an attempt to solve a rather interesting task, I encountered a problem. I need to solve the following exponential equations in natural numbers: $3^{b+1} = 2^{n+2} - 5$ and $3^{b+1} = 2^{n+2} + 1$. ...
Forsash's user avatar
0 votes
0 answers
21 views

Maximal abelian extension of $K^{ur}$

Let $p$ be a prime integer and $K$ be a $p$-adic field. For each integer $n\geq 1$, let $K_n/K$ be the unique unramified extension of degree $n$. Denote by $K^{ab}_n$ the maximal abelian extension of $...
Lam's user avatar
  • 53
5 votes
0 answers
215 views
+50

Find all the ordered triples of positive integers $(a,b,c)$ such that $b^c = 17+8^a$

Find all the ordered triples of positive integers $(a,b,c)$ such that $$b^c = 17+8^a$$ with $c>1$ I made $a=1$ and found that $(b,c) = (5,2)$ I made $a=2$ and found $(b,c) \in \{(9,2),(3,4)\}$ then ...
hellofriends's user avatar
  • 1,970
1 vote
0 answers
57 views

Periodic zeta function

Let $e(x)=e^{2\pi ix}$ and let $$F(x,s)=\sum _{n=1}^\infty \frac {e(nx)}{n^s}$$ be the periodic zeta function. What is the functional equation for the periodic zeta function ?: I can find a statement ...
tomos's user avatar
  • 1,662
0 votes
0 answers
113 views

Given a natural number $n$, is it always possible to find a prime number greater than $\frac{n}{2}$ but less than $n$?

Is the following statement true? If yes, prove it, else give a counterexample. Given a natural number $n$, is it always possible to find a prime number greater than $\frac{n}{2}$ but less than $n$? My ...
Olivia's user avatar
  • 851
1 vote
0 answers
258 views

7 for 5n+1 series diverges to infinity?

Write an odd integer in the "modified" binary form as $2^m - 1$ and apply 5n+1 (similar to applying the rules of original 3n + 1) to it: \begin{align} n &= 2^m - 1 \\ ...
Gaurav Goyal's user avatar
1 vote
2 answers
41 views

Fermat's last Theorem and elliptic curve cryptography

AFAIR, elliptic curve cryptography became popular soon after Fermat's last Theorem had been proven. Is it just a coincidence, or some important cryptographic properties of elliptic curves follow from ...
Roman Maltsev's user avatar
1 vote
0 answers
92 views

Determining rational solutions to $y^2 = 2(x^4 - 2x^3 - 2x - 1)$

I am trying to find all rational solutions to $y^2 = 2(x^4 - 2x^3 - 2x - 1)$ So far, I have tried to show that rational solutions to the above form an isomorphism with rational solutions to an ...
Ravikanth Athipatla's user avatar
4 votes
1 answer
136 views

2-adic valuations of $k\cdot 3^n-1$

I was just playing around with numbers of the form $3^n-1$, and noticed that their 2-adic valuations have a nice, understandable pattern: $\left(v_2(3^n-1)\right)_{n\ge 1} = (1,3,1,4,1,3,1,5,1,3,1,4,1,...
G Tony Jacobs's user avatar
1 vote
1 answer
55 views

$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?

I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function): $$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$ Over the interval $x=2$ to $x=10^4$ the average ...
zeta space's user avatar
0 votes
1 answer
38 views

A simple question about bounding a sum

Let $\Lambda_N:=\{-N+1,\dots,N-1\}^2\subset\mathbb{Z}^2$ be a set of lattice points in $\mathbb{Z}^2,$ and let $\gamma=\frac{1}{\sqrt{N}}.$ For $x\in\Lambda_N$ with $\|x\|_2>\gamma^{-1}$(with $\|\...
Chang's user avatar
  • 329
-1 votes
0 answers
35 views

Relation between the field and $\mathbb{Z}$-algebra generated by eigenvalues of modular form

Cross posted to MO: https://mathoverflow.net/questions/475273/relation-between-the-field-and-mathbbz-algebra-generated-by-eigenvalues-of Let $f$ be a cusp form of weight $k\in\mathbb{Z}$ for the group ...
1.414212's user avatar
  • 283
5 votes
1 answer
196 views

Conjectures involving $\Lambda(n)$

As the title suggests, I am looking for conjectures involving the Von Mangoldt function, $\Lambda(n)$. I understand this is not a rigorous mathematical question, however if reference requests for ...
Mako's user avatar
  • 702
0 votes
0 answers
35 views

Average cluster size of a nxn matrix

I asked a question about the cluster size inside a vector here. As a result, I finally used the expression $\frac{n}{-k+n+1}$ as the average cluster size, although it´s not proved correct for every ...
Cardstdani's user avatar
2 votes
1 answer
44 views

Ramification for a Galois extension of number fields

Let $f\in {\mathbb Z}[x]$ be a polynomial with discriminant $d\in {\mathbb Z}$. Let $L/{\mathbb Q}$ be the Galois extension obtained by adjoining to ${\mathbb Q}$ all roots of $f$. Claim. For all ...
Mikhail Borovoi's user avatar
-1 votes
0 answers
53 views

Are there infinitely many primes p for which either $p−1$ or $p+1$ is squarefree?" [duplicate]

It is not known if there is an infinite number of primorial primes. Are there infinitely many primes $p$ for which either $p−1$ or $p+1$ is squarefree?" I imagine this is an open problem also,...
Adam Rubinson's user avatar
2 votes
0 answers
72 views

Proving that sine is irrational at rational arguments with infinite fractions

In his proof of irrationality of $\pi$, Lambert uses the following continued fraction for tangent: $$\tan x=\cfrac{x}{1- \cfrac{x^2}{3-\cfrac{x^2}{5-\cfrac{x^2}{7-\cdots}}}}$$ He notes that for any ...
Loading - 146 Complete's user avatar
-1 votes
1 answer
75 views

Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."

I seek a highly detailed proof of this statement in the split multiplicative reduction case. The result can be found on page 57 here. That is, if $E/\mathbb Q$ has split multiplicative reduction at $p$...
Johnny Apple's user avatar
  • 4,429
-1 votes
1 answer
141 views

Is $\frac{a^{p}-1}{a-1}$ ($p$ prime) square-free? [closed]

$\frac{a^{p}-1}{a-1}$ should be square-free with all natural $a$. I've been looking for a while but no one online seems to have an definitive answer.
kiet's user avatar
  • 19
5 votes
1 answer
74 views

Finiteness of a generalization of the class number of a number field

Let $f(x)$ be an irreducible monic integral polynomial with root $\alpha \in \mathbb{C}$. A classical result of Latimer and MacDuffee asserts that there is a bijection between similarity classes of ...
Ben Marlin's user avatar
0 votes
0 answers
49 views

Mean of probability distribution

I have a probability distribution defined by the following density function: $f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
Cardstdani's user avatar
1 vote
0 answers
29 views

Proof of Grothendieck's Semistability Criterion for Elliptic Curves

We have the following result: Let $E/\mathbb Q$ be an elliptic curve and let $p$ be a prime. Then $E$ is semistable at $p$ if and only if $\rho_{E,\ell}|_{I_p}$ is unipotent for some (all) prime(s) $\...
Johnny Apple's user avatar
  • 4,429
1 vote
2 answers
60 views

Is it true that $A[2]\cong \varinjlim_i (A_i[2])$ if $A\cong \varinjlim_{i\in I} A_i$?

Let $I$ be a directed set. Let consider the direct limit in the category of abelian groups. Suppose $A\cong \varinjlim_{i\in I} A_i$. Then, is it true that $A[2]\cong \varinjlim_i (A_i[2])$ ? Here, $[...
Poitou-Tate's user avatar
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