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Results tagged with nt.number-theory
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user 39495
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
6
votes
Accepted
combinatorics on cyclic sequences
Answer: No this is not true.
For $m=5$, a counter example is: $(0, 2, 0, 2, 0, 2, 1, 2, 1, 1, 1, 0, 1, 2, 0)$. In this case we have:
$$\begin{align}
U(1) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
0 …
- 10.7k
3
votes
How many possible values for the determinant of an $n\times n$-matrix with entries $1,2,\dot...
I answer because Federico Poloni suggested it in a comment.
This sequence is in the oeis: oeis.org/A088217
I don't think the question is very well defined, what does "explicit function of $n$" mean? A …
- 10.7k
9
votes
Does $\{\tau(1)\tau(2)+\cdots+\tau(n-1)\tau(n)+\tau(n)\tau(1):\ \tau\in S_n\}$ contain a uni...
No, for $n = 11$ this fails:
363 = 3 * 11^2 with [7, 2, 8, 5, 3, 4, 6, 9, 10, 1, 11]
484 = 2^2 * 11^2 with [10, 9, 6, 3, 1, 2, 4, 5, 7, 8,
11]
Running the code I wrote to check this a little more, …
- 10.7k
12
votes
Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfe...
This is a to long for a comment:
Let $G(n,N)$ Micah's graph with vertices the numbers $1,..,n$ and edges $\{i,j\}$ if $i+j$ is a power of $N$. Your condition is satisfied if and only if $G$ contains …
- 10.7k
116
votes
Zagier's one-sentence proof of a theorem of Fermat
Let me answer your question "where do these involution come from" with an elementary geometric explanation. You can skip ahead to the pictures, which are somewhat self-explanatory, I hope.
The eleme …
- 10.7k
1
vote
Permutations $\pi\in S_{p-1}$ with $\frac1{\pi(1)\pi(2)}+\frac1{\pi(2)\pi(3)}+\cdots+\frac1{...
Not a proof, but some numerical evidence and an alternative conjecture:
It is straightforward to search permutations that do the trick for each $p$. When doing that for all $5\leq p\leq101$ I have fou …
- 10.7k
8
votes
0
answers
328
views
Asymptotics of A261668
In Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, Proposition 10.8, Jianqiang Zhao mentiones the sequence:
$$a_n=\sum_ …
- 10.7k
12
votes
1
answer
559
views
reference request: rational points on the unit sphere
I wonder what would be a good/early reference for the fact:
rational points on the unit sphere (centered at the origin) are dense.
Stereographic projection (from a rational point in the sphere) …
- 10.7k