My claim:
Any set of exactly $n$ integers in the range $[1,n^2]$ which are an arithmetic sequence and share no common factor will contain at least one prime.
Let $n\geq 2$ be a natural number, and let $a,b\in\mathbb{N}$ where $1\leq a \leq n$ and $1 \leq b \leq n^2$, such that $an+b\leq n^2$. Then the set $\{ax+b \mid 1 \leq x \leq n\}$ will contain between $1$ and $n$ primes, inclusive.
There are three exceptions: $(n=12,a=1,b=113),(n=12,a=1,b=114),(n=13,a=1,b=113)$, but I strongly suspect those are the only ones. (You could eliminate them by requiring $a > 1$, but I don't think it's worth the cost in generality.)
In a nutshell, this conjecture claims that given an $n \times n$ square, any line you draw (that intersects the integers appropriately) must contain a prime, including for starters all rows and columns—something I called the Rook Conjecture, but have since seen referred to as the Calendar Conjecture.
I've empirically tested this out to $n=150$ or so, and the trend is strong enough that I'm almost sure it's correct. Of course, finding a proof is another story.
I'm looking for feedback. Good answers would be:
- counterexamples
- indicating that this is previously known and/or conjectured
- providing reasons why this does or does not seem likely to be true
- any other insights