posted on Sep, 24 2009 @ 10:41 AM
Originally posted by mathfreak
Finding all the squares of primes (except 4 and 9) on line 1 is really cool, something I didn't know, and I salute you for it, but there's a reason
for this, too. Modular arithmetic says we only need to check the squares of the first circle. If they all lie on line 1, then all possible prime
squares will lie on line 1.
It's pretty easy to see why. Consider "p^2 - 1", where "p" is prime. The value will always be an even even multiple of 24. This is because
"p^2 - 1" can be factored as "(p-1) * (p+1)".
Consider
any sequence of three contiguous integers: "(x-1), x, (x+1)".
Two of them
must be divisible by two (even numbers). One of them must be divisible by three, and one of them
must be divisible
by four. In the case of a prime number "p", which of course is not divisible by anything, then we can say: one of (p-1) or (p+1) is divisible by
four, the other by two. Additionally, one of them is divisible by three.
Thus, "p^2 - 1" always has the factors 2, 4, and 3, and is divisible by 2*4*3 = 24. That's why it's always on the same radial line.