[phillip1882]

so i was bored at work and started to once again mess with my favorite sequence, the prime numbers.
i came up with the following interesting observations.
the ratio of the number of prime numbers between n/2 and n to n and 2*n is roughly the golden ratio (1.618).
for a few examples, 300/2 = 150. the number of primes between 150 and 300 is 28.
300*2 = 600. the number of primes between 300 and 600 is 47.
28*1.618 = 45 aprox.
on a similar vien, ratio #p between n/phi and n to n and n*phi is roughly sqrt(2).
once again, letting n be 300,
300/phi = 185, #p between 185 and 300 = 20,
300*phi = 485, #p between 300 and 485 = 30.
20*sqrt(2) = 28 aprox.
the exact golden value for the primes seems to be roughly
1.489, or 1/3 +1/5 +1/5 +1/7 +1/11 +1/13 +1/17 +1/19 +1/29 +1/31 +1/41...
that is... ratio #p between N/1.489 and N to N and N*1.489 is roughly 1.489.
cool eh?

[CraigD]

phillip1882, on 14 February 2012 - 01:11 PM, said:

... cool eh?

Very!

As a fellow spelunker of the primes, my first thought was this has to be related to the prime number theorem, and the logarithmic integral function, li().

I didn’t get as far or cool as you, Phillip, but taking the li(n) - li(2) as a good approximation of the number of primes less than n, and multiplying by 2 to get rid of annoying fractions, you can write you maybe phi-ish ratio



There are lots of series for approximating li(), including ones built into calculators like Wolfram Alpha, so I could quickly calculate these approximate values of it:

x		(Li(4x)-Li(2x))/(Li(2x)-Li(x))
100		1.75549
1000		1.82621
10000		1.86517
100000		1.88985
1000000		1.90689
10000000	1.91937
100000000	1.92889
1000000000	1.93641
10000000000	1.94248
100000000000	1.9475
1000000000000	1.95171
10000000000000	1.9553
100000000000000	1.95839

They look to me not to be trending toward 1.618, only to be near it for small prime counts. There might be a limit, but that’s just a guess.

Since li() has simple infinite series representations, I bet one could work out/prove exactly both of your ratio conjectures.