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Topic: Prime numbers and pi
Replies: 33   Last Post: Dec 10, 2012 3:43 PM

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 Franz Gnaedinger Posts: 330 Registered: 4/30/07
Re: Prime numbers and pi
Posted: Apr 16, 2009 11:06 PM

The natural logarithm of 2 as an infinite series and an
infinite product, arranged in pairs of terms:

ln2 = '1x2 '3x4 plus '5x6 '7x9 plus '10x11 '13x15 plus ...

ln2 = 1x3/2x2 times 5x7/6x6 times 9x11/10x10 times ...

The average of the first terms is 0.625, the one of the
first two terms 0.68184..., the one of the first three
terms 0.68725..., and so on. The average of the ever
longer partial series and the ever longer partial product
approximates ln2.

Euler's pi involving the primes, in a modified form:

pi/4 = 1x3/2x2 times 5x7/6x6 times 11x13/10x14 times ...

Another infinite pi product involving the numbers
1 4 7 10 13 16 19 22 25 ...:

2/pi = 1x4/2x3 times 7x10/8x9 times 13x16/14x15 times ...

The product of the two infinite products equals 1/2.
Multiply the first terms and you get 1/2. Multiply the
subsequent terms, the first terms, the second terms,
the third terms ... (only the terms, not the ever longer
partial products), and the values oscillate around 1.

May it be that the numbers 1 4 7 10 13 16 19 22 25 ...
are in some mysterious way related to the primes?

'1x2 '3x4 '5x6 '7x8 '9x10 ... = ln2

'1x3 '5x7 '9x11 '13x15 '17x19 ... = pi/8

'1x4 '7x10 '13x16 '19x22 '25x29 ... = ???