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Re: Prime numbers and pi
Posted:
Apr 16, 2009 8:33 AM
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Stairway approximations of ln2 :
'1
'1x2 '5
'1x2 '3x4 '9
'1x2 '3x4 '5x6 '14
'1x2 '3x4 '5x6 '7x8 '19
'1x2 '3x4 '5x6 '7x8 '9x10 '24
ln2 = '1x2 '3x4 '5x6 '7x8 '9x10 '11x12 '13x14 '15x16 ...
9/10
1x3/2x2 x 14/15
1x3/2x2 x 5x7/6x6 x 19/20
1x3/2x2 x 5x7/6x6 x 9x11/10x10 x 24/25
ln2 = 1x3/2x2 x 5x7/6x6 x 9x11/10x10 x 13x15/14x14 ...
ln2 = 1/2 x 3/2 x 5/6 x 7/6 x 9/10 x 11/10 x 13/14 ...
By playing with this product and looking out for
partial products Euler could have left out all factors
whose numerators are composites and would have obtained
1/2 x 3/2 x 5/6 x 7/6 x 11/10 x 13/14 ... = pi/4
At least half an explanation for that amazing link
between pi and the primes.
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