# Tag Info

Accepted

### A proof of Van der Waerden's theorem using a weakened form of Szemeredi's theorem

As (implicitly) observed already in Szemerédi's celebrated paper Szemerédi, Endre, On sets of integers containing no (k) elements in arithmetic progression, Acta Arith. 27, 199-245 (1975). ZBL0303....
• 91.7k
Accepted

### What is the status on this conjecture on arithmetic progressions of primes?

Yes, this is unknown; it is even unknown (as GH from MO suspected in a comment) whether $P(p) \ge 3$ always. An equivalent statement to $P(p) \ge 3$ is that there exists an integer $x>0$ such $p+x$ ...
• 12.4k
Accepted

• 9,059
Accepted

### Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

You are essentially asking for quantitative estimates on Szemerédi's theorem, which states that the largest subset of $[1,n]$ without a k-term arithmetic progression has size $o(n)$. To be precise, ...
• 756
Accepted

### Arbitrarily long arithmetic progressions

A simple proof is available as well. Pick p coprime to d and let t be such that td=1 mod p. Then, mod p, t times the arithmetic progression looks like a sequence of consecutive integers. Thus its ...
• 2,108

### Primes from a Dirichlet sequence and an irrational number

It is well-known that not only does the arithmetic progression $\{ak+b\}_{k \in \mathbb{Z}^{+}}$ contain infinitely many prime numbers, but also that the series of the reciprocals of those primes ...
• 8,197

### Is there an 11-term arithmetic progression of primes beginning with 11?

Siemion Fajtlowicz has been promoting the topic of $p$-long arithmetic progressions of primes which start with $p$ during 1993-4 (or longer). He and his colleague Micha Hofri got an $11$-long ...
• 3,655

### Largeness and arithmetic progression properties of generic reals

Nice question! I happened to be thinking about some similar things a few weeks ago. Here is what I found: Cohen and random reals: Cohen and random reals have just about any Ramsey-theoretic property ...
• 15k

### Arbitrarily long arithmetic progressions

If $x,y,z$ are in arithmetic progression, then $x+z-2y=0$. By the S-unit theorem of Evertse, Schmidt and Schlikewei, this equation has only finitely many solutions in $x,y,z$ having all its prime ...
• 29.6k

### Smallest set such that all arithmetic progression will always contain at least a number in a set

Considering the complement of $P$ in $[1,100]$, you are asking how large can a subset of $[1,100]$ be given that it does not contain any $10$-term arithmetic progression. The more general question ...
• 21.8k
Accepted

### A set with positive upper density whose difference set does not contain an infinite arithmetic progression

Let $\langle x\rangle$ denote the fractional part of a real number $x$ (i.e. $\langle x \rangle := x- \lfloor x\rfloor$, where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$). ...

### Smallest set such that all arithmetic progression will always contain at least a number in a set

Using a tabu search procedure, I have found a solution for $|P|=17$, namely ${1, 11, 18, 25, 31, 32, 33, 36, 44, 51, 58, 65, 69, 70, 77, 84, 91}$. I don't know if this is optimal. EDIT: Found a ...
• 51.7k