I was messing around in Excel at the end of work today and made a table where the $(i,j)$ entry $a_{i,j}$, for $j \geq i$, is 1 exactly when $i$ and $j$ are coprime (see snapshot of a portion of the table below):
My questions are:
- Is there an explanation for the highlighted rectangular pattern that keeps popping up (albeit in different orientations)?
- What about the numbers where $a_{*j}$ is $0$ for "most" values of $j$. They seem to be sandwiched between twin primes or at least right next to a prime. Is there any significance there?
- Another question that popped out of this, are there infinitely many solutions for $\phi(n) = \phi(n+1)$? I also calculated $\phi(n)$ directly at the top of my table and noticed that the pattern for solutions seemed somewhat erratic. Is there a theorem related to this at all?
These questions are rather open ended and I apologize. Just trying to see if there is any merit to my curiosity today.
Twin primes are always either side of a multiple of 6 (except for the first twin primes, 3 and 5). Why is that?
I think the numbers with most blank space below them are multiples of 6. Why would most numbers have a factor in common with any multiple of 6?
I don't know about your last question. None the less, playing around with maths is (a) fun and (b) how a lot of maths gets discovered.