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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
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 …
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 …
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 …
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 …
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 …
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, …