All Questions
40
questions
16
votes
1
answer
986
views
Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle as a combinatorics problem
It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be ...
- 10.5k
14
votes
0
answers
288
views
All interval sequences mod integers
In music, an all-interval twelve-tone sequence is a sequence that contains a row of 12 distinct notes such that it contains one instance of each interval within the octave, 1 through 11. The more ...
- 2,733
10
votes
5
answers
211
views
$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view
A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity
\begin{align*}
\zeta(2,1)=\...
- 109k
10
votes
0
answers
357
views
Dissecting the complexity of prime numbers
Each prime number greater than $9$, written in base $10$, ends with one of the four digits $1,3,7,9$. Therefore, each ten can be classified according to which of these four digits, summed to the ten, ...
user559615
8
votes
2
answers
168
views
Is there any elementary solution for this problem on colored interval?
The problem is as following.
Assume $m,n$ are two coprime odd numbers, consider the interval $[0,mn]$. We cut the interval by $m,2m,\ldots,(n-1)m$ and $n, 2n,\ldots, (m-1)n$ into $m+n-1$ pieces of ...
- 1,592
7
votes
2
answers
199
views
Are there any non-casework proofs of the $18$-point problem?
There is a cute little problem stated as follows:
Choose a sequence $x_1,x_2,x_3,\ldots$ with $x_i\in[0,1)$ such that $x_1$ and $x_2$ are in different halves of the unit interval, $x_1,x_2,$ and $x_3$...
- 17.7k
7
votes
1
answer
398
views
"On the consequences of an exact de Bruijn Function", or "If Ramanujan had more time..."
In this question on Math.SE, I asked about Ramanujan's (ridiculously close) approximation for counting the number of 3-smooth integers less than or equal to a given positive integer $N$, namely,
\...
- 17.1k
7
votes
1
answer
175
views
Gowers' proof of Szemerdi's theorem
Are there any good books or other resources (expository notes) which explains Gowers' proof of Szemerdi's theorem in detail?
- 15.1k
5
votes
1
answer
460
views
Arithmetic progressions
What are the largest known lower bounds for $B_k$, the maximal sum of the reciprocals of the members of subsets of the positive integers which contain no arithmetic progressions of length $k$?
for $k=...
- 53
5
votes
1
answer
218
views
Pretty $p^2$-congruences involving Stirling numbers of the both kinds
Let $p$ an odd prime number and ${n\brack {k}}$ (resp. ${n\brace k}$) be the Stirling numbers of first (resp. second) kind, such that:
$$ \sum_{k\ge0} {{n}\brack {k}}x^k = \prod_{j=0}^{n-1}(x+j)$$
$$ ...
- 3,716
5
votes
0
answers
2k
views
Good books to learn olympiad geometry,number theory, combinatorics and more
I want to start learning olympiad mathematics more seriously, and I would like to have advice on some good books or pdfs to learn with.
I have background but not a big background. For example I know ...
- 2,510
4
votes
3
answers
1k
views
Largest subset with no arithmetic progression
I am trying to find some weak bounds on the largest subset of a set, such that the subset has the property that it contains no three elements in arithmetic progression. The elements of the original ...
- 2,242
4
votes
1
answer
149
views
FLT (Fermat): Combinatorial approaches?
Such a simple equation like $x^n+y^n=z^n$ is bound to have a nice/natural combinatorial interpretation. One very crude one is: Let the number of ways of choosing $n$ objects from $x$ objective, ...
- 2,242
4
votes
1
answer
251
views
Novel(?) method of generating Motzkin numbers, Catalan partial sums, and other sequences
For solutions to the equations (1/X)+(1/Y)=K and (X+Y)=1+(1/k) the answer is a pair of points. The X and Y values of a point represent stable points in a two-cycle orbit of the logistic equation ...
- 41
3
votes
1
answer
99
views
Sum of products of $m$-tuples chosen from the set of squared reciprocals
Let $S = \{1/n^2 : n \in \mathbb{N} \}$. We know $\sum S = \zeta(2) = \pi^2/ 6$.
Let $f(S, m)$ be the sum of the products of all $m$-tuples chosen from $S$. That is
$$f(S,m) = \sum_{X \in {S \...
- 10.9k