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 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.
71 questions from the last 30 days
3
votes
1
answer
75
views
Prove $\frac{x^{2^n}-1}{x^{2^m}-1}$ is not a perfect square when $n, m$ have different parity
I am trying to prove the following number theory problem:
Problem:
Let $n, m$ be positive integers with different parity (one is even, the other is odd) and $n > m$. Prove that there is no integer $...
1
vote
1
answer
57
views
Does there exist a prime decomposition for every possible combination of ramification and inertia degree
I just finished the treatment of quadratic fields and cyclotomic fields in Marcus' Number Fields and I decided to approach biquadratic fields as a fun exercise. From the Ram-Rel identity we know that ...
- 193
3
votes
1
answer
416
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Why elliptic curves instead of Edwards curves?
I have been wondering this for a while; I get how elliptic curves of the Weierstrass form $y^2=4x^3-g_2x-g_3$ have the lowest degree and are the simplest way to study a genus $1$ curve. But why aren’t ...
- 360
4
votes
1
answer
60
views
Minimum cardinality of the set of values for a sequence($a_1,a_2...a_{2025}$) with distinct cyclic ratios
Let $n = 2025$. We are given a sequence of positive integers $a_1, a_2, \dots, a_n$.
Let the cyclic ratios be defined as:
$$r_i = \frac{a_i}{a_{i+1}} \quad \text{for } 1 \le i \le n-1, \quad \text{and}...
2
votes
0
answers
62
views
Which permutations Galois considered out of $n!$ permutations in $S_n$ to prove insolvability of quintic polynomial?
It seems to me that the beginning of Galois theory (Galois's work) is still not clear to me. To prove the quintic or higher degree polynomial is not solvable in radicals, Evertise Galois considered ...
- 574
1
vote
0
answers
45
views
A conjecture on representing composite Fibonacci numbers
I've been exploring a question about composite Fibonacci numbers and I'm not sure if my findings are new or if this is a known problem.
Definitions
Let $F_n$ be the $n$-th Fibonacci number ($F_1=1, ...
5
votes
1
answer
123
views
A super-exponential sequence related to the Fibonacci sequence
Consider the sequence $(x_n)_{n\ge 0}$ defined by
$$x_0=1,\;\;\;x_1=1,$$
and for $n\ge 1$
$$x_{n+1}=\sum_{0\le i\lt j\le n}x_ix_j\;\;\;\;\;\;(1)$$
So $x_{n+1}$ is the sum of all pairwise products of ...
- 3,054
0
votes
1
answer
67
views
A “block” generalization of factorions
I’ve been exploring a generalization of factorions—that is, numbers equal to the sum of factorials of their digits—and wanted some feedback on the concept.
Let $n \ge 1$ where $n$ is an integer. A ...
- 13
0
votes
2
answers
113
views
Is there a subset of the positive integers with this property?
This question was partly inspired by my previous one, here: A strengthening of the Green-Tao theorem. As in that question, I will restate the definition of a maximal arithmetic progression. Let $k$ be ...
- 24.8k
0
votes
1
answer
109
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Solutions to $x^n \equiv x \pmod{n}$ for composite $n$.
By Fermat's Little Theorem we know that if $p$ is some prime number, the congruence $x^p \equiv x \pmod{p}$ is solved by any integer $x$, but can we say something about the solutions to $x^n \equiv x \...
- 7,746
-3
votes
0
answers
59
views
Euclid's Labour [closed]
edit: projecteuler.net discourages sharing direct solutions for problems after 1-100, i just need some hints, then i can work up from there
problem 958 (Euclid's Labour) from projecteuler.net (...
- 1
0
votes
0
answers
62
views
What is the least possible value of p+q? [duplicate]
Problem: $$n^{3pq}- n$$ is a multiple of $3pq$ for all positive integers n. Find the least possible value of p + q?
My Progress:
If $$ n^{3pq} \equiv n \pmod{3pq} $$ for all n, it hints at using ...
- 79
2
votes
1
answer
99
views
Inequality involving Euler's totient function on three consecutive integers
I am considering the following inequality involving Euler’s totient function $\varphi$:
For every integer $x>4$,
$$
x \le \varphi(x-3) + \varphi(x-2) + \varphi(x-1).
$$
$x$ from $5$ to $200{,}000$ ...
- 29
3
votes
1
answer
75
views
Continuity of a function under divisor topology
Let $X= \{ 2,3,\dots\} \subset \mathbb N$. We define the divisor topology on $X$ by establishing a basis $\mathcal B = \{S_n \}_{n \in X}$ where $$S_n = \{p \in X : p|n\}$$
($p$ divides $n$).\
Under ...
- 311
2
votes
0
answers
63
views
Copeland-Erdős constant random walk
WOW! Here are two photos of Copeland-Erdős constant with first 500 digits and first 4000 digits. In Copeland-Erdős constant all prime numbers are presented in order in decimal so $0....
