I have a really crappy English class. We're doing a unit on public speaking, and we get to choose a topic and give a "lecture" to the class. Some people are doing instructional talks, like how to tie shoelaces; and other people are giving persuasive talks.
To annoy the crap out of my teacher, I'm doing a 45 minute long presentation on the Green-Tao Theorem. If you have any suggestions or links to good papers, I'd appreciate it, but that's not what this note is about.
I have discovered a truly marvelous property that may help with a compact proof. I observed (in my science class) that given an arithmetic sequence, there tended (see the last paragraph for more info) to be prime numbers in prime positions! For example, consider the sequence:
Notice how all boxed numbers are primes. Here is a list of the positions at which the prime numbers appear:
These are all primes! In fact, these are all CONSECUTIVE primes! Wow! Why is this? Can we generalize this for any arithmetic progression ? Is it a coincidence that the difference in this sequence () is prime, since the same property obviously wouldn't hold for ? And is this an application of the GT Theorem, or part of its proof? Isn't it just the difference in the APs that matter because can be shifted back or forward by to align the primes into prime positions?
I've taken steps toward a proof that may or may not be really awesome. I'll publish it when (or if) I finish it. What do you think?
As another example, consider , . The sequence is:
which has primes at positions:
which seems pretty odd (get it?).
As pointed out by Daniel Liu, each "good" sequence such as the first one () can be shifted by changing . This will throw the "prime pattern". But what needs to be shown is that there exists an optimal for ALL . In this case, at the optimal solution is . What about ? What can we say about ? Must it be prime? What else can be observed?