Prime pattern

nate808

I am curious as to whether this pattern will always hold true:

Let's say we take the prime numbers:
2,3,5,7,11,13,17,19,23.......primes
and we take the square(individually) minus 1
3,8,24,48,120,168,288,360,528....p^2 - 1

Then starting with the third p^2 - 1 (24), all of the p^2 - 1 can be rewritten as and earlier p^2-1 times a prime # (or multiple prime #'s)

for example: 24=8x3,48=24x2,528=48x11,...and so on.

Is this at all interseting or just something stupid that I am missing?

Let me know,
Thanks

Hurkyl

Why not just say "p²-1 times a number"?


It's a rather trivial observation; 3 is on your list.

nate808

The number 4 times out of five is a single prime #. It seemed a lot nicer when i posted it and in the first ten i checked 9 of them were single prime numbers. I have since found 3 more where the # is a multiple of 2 primes (2 and 3 in all cases). Thanks for at least looking at it Hurkyl

Hurkyl

I suspect what you're seeing is simply a property of small numbers, rather than anything special about your construction. That's why I asked how far you looked -- I imagine when you start looking at numbers in the trillions, you won't see this sort of thing very frequently at all.

matt grime

Since you can factor p^2-1 you should start to see that this is not likely to be a recurrent pattern (an even weaker hypothesis is for every prime p, either p-1 or p+1 to have a repeated factor). This seems highly unlikely to be true in general.