0
votes
0answers
35 views

What is the explanation for the $64$ in Graham's number $g_{64}$?

As in, why does the iteration of the function until $g_{64}$ guarantee this property that defines Graham's number? Why was this number chosen? If I had to guess (emphasis on guess), I'd say that the ...
2
votes
1answer
68 views

A problem relying on van der Waerden's theorem, and the existence of sums divisible by a given number $n$

Say we are given a sequence of integers $\{a_i\}_{i \in \mathbb{N}}$, as well as a pair of integers $n, m$. How can we show that there always exist positive integers $s, r$ such that the sums $a_s + ...
2
votes
0answers
83 views

Van der Waerden type numbers (for geometric progressions)

Van der Waerden theorem is true also for geometric progressions. Is there anything interesting in van der Waerden type numbers $ W'(r,k) $ derived from this version? ($ W'(r,k) $ is such that if the ...
4
votes
1answer
144 views

A problem in prime number theory

I was wondering if anybody here might provide me with a hint for this rather innocuous-looking problem: If $X:= \{pq: p, q \mbox{ are prime numbers and } p\neq q\}.$ In addition, let us suppose that ...
6
votes
2answers
448 views

Proof of the van der Waerden theorem

The van der Waerden theorem states that given any natural numbers $k$ and $r$, there exists a natural number $W=W(k,r)$ such that if the set $\{1,2\cdots W\}$ is divided into $r$ classes (also called ...
3
votes
1answer
126 views

Van Der Waerden without topological dynamics?

Applying topological dynamics to prove Van Der Waerden's theorem on the existence of monochromatic arithematic progression has now become a somewhat classical example of the power of topological ...