for prime number-fags: I'm trying, just for fun, to figure out how to prove the following property of prime numbers: given three consecutive prime numbers x, y, z, the bigger they are the more their sum x\y + y\z + z\x tends to be 6. any suggestion?
It is very easy to prove it false. Take three random consecutive primes and test those ratios. They will be very far away from six.
Then take another three bigger consecutive primes. They will be even more far away from six. Theorem discarded.
x < y < z
x\y + y\z + z\x
the first two are smaller than 1 and the last is quite big.
I don't see how this would converge to something small.
Pretty sure this is correct
(I think) You can use the prime number theorem's estimate of ln(x) for prime gaps as you go to infinity,
x/y -> x/(x+lnx) -> 1
y/z -> y/(y+lny) -> 1
z/x -> (y+lny)/x = (x+lnx+ln(x+lnx))/x -> 1
The prime numbers are quite common :-).
If I remember correctly about (n / ln n) in 1..n (n -> inf).
So three consecutive primes will be really close (relatively). As they get bigger the sum will get closer and closer to 3.
This works for any number.
When the numbers are EXTREMELY HUGE and close together, they're essentially the same number, very small difference.
And dividing a number by something that is basically the same number, you get 1
So its literally just 1+1+1=3
OP, you are a brainlet if this was surprising to you.