# 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 ...
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### 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 ...
• 69
1 vote
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### 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 $\frac{\deg{f} - 2}{2}$, ...
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 ...
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1 vote
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### 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) ...
• 5,633
1 vote
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 ...
• 32.3k
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### 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 ...
• 329
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 ...
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1 vote
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### 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 ...
• 21
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### 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 ...
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### 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 ...
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 ...
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1 vote
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### 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 ...
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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 ...
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 $3^{b+1} = 2^{n+2} + 1$. ...
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### $H(x)$ 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 ...
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### 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, \$[...