Recently I saw a video on YouTube where the Fibonacci numbers were studied and around minute 4:20 appears a graph showing the period against the modulus. Something that caught my attention is that when plotting many points straight lines begin to appear whose slopes seem to follow some kind of relationship (see OEIS A001175).
When studying other sequences that appear in The On-Line Encyclopedia of Integer Sequences (OEIS) I found this same behavior (for example: OEIS A175181, OEIS A060305, OEIS A253246). On my own I obtained another graph, putting on the X-axis prime numbers and on the Y-axis their Pisano periods, and I obtained the emergent behavior of lines appearing when plotting many points:
Pisano periods of prime numbers
My first question is, why do these lines appear? is there any concise mathematical explanation behind their origin?
Comparing the plot I obtained with the plot shown in the video, I observe that the lines using only primes are much more "defined", in the sense that they can be better distinguished from each other. Do you know if there is an explanation for this behavior? Could it be that the prime numbers have some interesting property with respect to their Pisano period?
Finally I plotted the slopes of the lines using only prime numbers with respect to the line number (I am counting them from top to bottom), and I am getting that they follow a kind of power law: $y\sim 2/x$
Power law fit for the slopes obtained in the previous plot
Do you know if there is any mathematical explanation of why it follows this relationship?
I thank you in advance for all your answers,
Alex