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Results tagged with nt.number-theory
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user 120914
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
4
votes
Compute generators of $\Gamma_0(N)$
sage should have what you need.
Check under the documentation for the modular group. Specifically under generators you can find a couple of working examples for finding the generating set for $\Gamm …
4
votes
Accepted
What are the exceptional properties of Mersenne exponent for known largest prime?
Primes $p$ such that $2^p-1$ is prime are called Mersenne exponents. You can find many of their properties on the corresponding entry of the OEIS, namely A000043. Beyond their obvious connections with …
3
votes
Accepted
Behavior of biggest prime divisor of $n$ as $n$ grows large
The number of prime divisors of $n$ grows typically as $\log \log n$. Suppose $n$ has $k$ prime factors. Now $n/L(n)$ has only $k-1$ prime factors, so
$$
k-1 \approx \log \log \frac{n}{L(n)} = \log \l …
8
votes
1
answer
650
views
Arithmetic sequences and Artin's conjecture
(Sorry if this is a naive question; it is not my area!)
Consider the following strengthening of Artin's conjecture on primitive roots (and Dirichlet's theorem) for the case of $n=2$: every arithmetic …
15
votes
Accepted
Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function
(Expanding my comment into an answer)
It is not a coincidence. Relating the Euler characteristic of certain arithmetic groups to the Zeta function is a theorem due to Harder [1] from 1971. It is expan …
2
votes
Accepted
How to calculate genus number of number field using sage?
I may as well promote my comment to an answer, so this is closed. In a short paper, H. Hasse [1] computed the genus field and genus number of every real quadratic field. In particular, he shows that, …
25
votes
Why is integer factoring hard while determining whether an integer is prime easy?
Let me give a slightly different example. The Baumslag-Gersten group is a one-relator group with the presentation $\langle a, b \mid [a, a^b] = a\rangle$. This has a Dehn function which grows faster t …
5
votes
Accepted
Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathb...
This is a hard problem in general. When $N$ is not prime, then even the first homology of $\operatorname{PSL}_2(\mathbf{Z}[\frac{1}{N}])$ is non-trivial to compute (cf. Corollary 4.4 of [1]), but I re …