Unanswered Questions
7,492 questions with no upvoted or accepted answers
138
votes
0
answers
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Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 2,758
78
votes
0
answers
3k
views
The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p)...
- 4,745
67
votes
0
answers
3k
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Constructing non-torsion rational points (over $\mathbb{Q}$) on elliptic curves of rank $>1$
Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
- 19.3k
58
votes
0
answers
4k
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Grothendieck's Period Conjecture and the missing p-adic Hodge Theories
Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
- 69
55
votes
0
answers
3k
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On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
- 2,598
51
votes
0
answers
2k
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Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?
A very specific case of Reed's Conjecture
Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...
CommunityBot
- 1
45
votes
0
answers
3k
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A = B (but not quite); 3-d arrays with multiple recurrences
Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan)
1
1 3
2 10 15
6 40 105 105
24 196 700 1260 945
...
- 33.4k
43
votes
0
answers
1k
views
Can a regular icosahedron contain a rational point on each face?
The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces?
For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, ...
- 12.7k
41
votes
0
answers
1k
views
Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?
Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
- 62.6k
41
votes
0
answers
2k
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What does the theta divisor of a number field know about its arithmetic?
This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let ...
CommunityBot
- 1
40
votes
1
answer
5k
views
A generalisation of the equation $n = ab + ac + bc$
In a result I am currently studying (completely unrelated to number theory), I had to examine the solvability of the equation $n = ab+ac+bc$ where $n,a,b,c$ are positive integers $0 < a < b < ...
- 62.2k
36
votes
0
answers
2k
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3-colorings of the unit distance graph of $\Bbb R^3$
Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = (\alpha,...
- 16.8k
35
votes
0
answers
945
views
Orthogonal vectors with entries from $\{-1,0,1\}$
Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $...
- 5,952
35
votes
0
answers
1k
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Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?
The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
- 62.6k
34
votes
0
answers
1k
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Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?
In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.
QUESTION Is there any (added: ...
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