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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Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$?
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 ...
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Why do Fibonacci sequences result from this process?
If we have two columns of numbers made by the following rule, we get two Fibonacci sequences.
Is there a straightforward way that enables us to just 'see' why this would happen. If anyone can find ...
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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 $\frac{\deg{f} - 2}{2}$, ...
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$a^3 + b^3 + c^3 = 4abc$ 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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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.
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{m=...
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Prime-Independence of p-adic Continued Fractions: New Observation?
I've recently been exploring p-adic continued fractions and stumbled upon an intriguing pattern. It seems that the p-adic continued fraction representations of rational numbers are consistent across ...
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Analogue of roots of unity in n-sphere
The $n$-th roots of unity $z_1,…,z_n$ in $\mathbb{C}\equiv \mathbb{R}^2$ for $n$ prime have an interesting property: for $0\leq p<n$ and $u$ a unit vector, the sum $$\sum_{k=1}^n \langle z_k,u\...
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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) ...
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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 ...
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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 ...
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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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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)+...
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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 ...
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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 ...
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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 \...
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