27
votes
Accepted
Object of proven finiteness, yet with no algorithm discovered?
Most of Diophantine approximation falls into the category. For example, Roth's theorem says that for any non-rational algebraic number $\alpha\in\overline{\mathbb Q}\smallsetminus\mathbb Q$ and any $\...
- 46.7k
17
votes
Object of proven finiteness, yet with no algorithm discovered?
Heath-Brown proved that Artin's conjecture on primitive roots has at most two counterexamples. So for example, we know that the conjecture is true for at least one element of the set $\{2,3,5\}$, but ...
- 80.3k
11
votes
Object of proven finiteness, yet with no algorithm discovered?
Another example is Euler's idoneal number problem, one of the oldest problems in number theory. It asserts that $1848$ is the largest number $n$ such that the class group of the quadratic order $\...
- 25.9k
10
votes
Object of proven finiteness, yet with no algorithm discovered?
There exists a positive integer $N \le 246$ such that there are infinitely many primes $p$ for which $p + N$ is also prime. However, there is no single specific value of $N$ for which this is proven.
- 1,643
10
votes
Object of proven finiteness, yet with no algorithm discovered?
The graph minor theorem implies that any family of graphs that is closed under minors has a finite forbidden minor characterization, but the proof does not yield an algorithm for finding the minors. ...
- 80.3k
9
votes
Object of proven finiteness, yet with no algorithm discovered?
We know that the chromatic number of the plane is at most 7 but do not know its exact value. To remove any dependency on AC, we could ask for the maximum possible chromatic number of (the unit-...
- 2,848
6
votes
Object of proven finiteness, yet with no algorithm discovered?
$\DeclareMathOperator\BB{BB}$My answer is a slight cheat, because I'm going to give two examples where we have not only not discovered any algorithms but where we know there is no such algorithm. Both ...
- 6,819
5
votes
Accepted
Number of planes generated by integer vectors
For $k=d-1$ this is a result of Bárány-Harcos-Pach-Tardos (2001). See Theorem 3 in the preprint version or the published version.
- 102k
4
votes
Rate of convergence of the Riemann zeta function and the Euler product formula
I'll use a variable $m$ for the $m$th approximation in the sequence since the variable $n$ is in use.
On the left hand side, the $m$th approximation is $$\sum_{n=1}^{m} n^{-s}$$ with error $$\sum_{n=m+...
- 141k
4
votes
On an integer factoring algorithm based on smooth class number of quadratic fields
This idea was first studied by Shanks, Pollard, Atkin and Rickert, although they didn't write a paper as far as I know. Schnorr and Lenstra gave a heuristic time complexity analysis in their 1987 ...
- 191
3
votes
Object of proven finiteness, yet with no algorithm discovered?
Here is an example where finiteness is known, an algorithm exists, but such an algorithm is ineffective, in the sense that current computers (to my knowledge) won't terminate in any reasonable amount ...
2
votes
Minimum value of a function involving the divisor counting function
First note that if $t \not\mid n$, $f_n(\gcd(n, t)) < f_n(t)$.
We will split to cases:
If for the minimal $m$, $\gcd(m,n) \neq n$, we can assume $m \mid n$, otherwise $\gcd(n, m)$ would've been ...
- 3,049
2
votes
Possible new series for $\pi$
A related, and perhaps easier, question is whether there are other known series for 𝜋 that involve a complex parameter 𝜆 in the summand, but where the sum of the series is independent of the value ...
- 2,510
2
votes
Ramanujan type identity for level 3 and Weber function
The answer is no, this transformation does not exist. This is a Corollary of Theorem 2.2 in Mitsuo Kato article.
Kato's theorem proves the existence of algebraic transformations between $\,_3F_2$ ...
- 2,490
1
vote
Accepted
Proof that the infinity norm of basis matrix is greater than or equal to 1
For $\alpha=1$ we have that
$\max_j|a_j|\geq 1$, while $\max_k|\alpha^{(k)}|=1$. This proves that $\|B^{-1}\|_\infty\geq 1$.
- 102k
1
vote
Accepted
Finding a two point scrambled set for the function $g:[0,1] \rightarrow [0,1], x \mapsto \min_{n\in \mathbb{Z}} |3x-2n|$?
Having digested your definitions... Any number which has an aperiodic orbit (not a fixed point), and any number which has a periodic orbit (possibly a fixed point), will between them constitute a two-...
1
vote
Why are modular forms interesting?
For future reference: in Lizhen Ji's introduction (pp. xv–xvii) for DuPre's translation of Klein–Fricke, they address your question.
Klein, F., & Fricke, R. (2017). Lectures on the Theory of ...
Only top scored, non community-wiki answers of a minimum length are eligible