Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Franz Gnaedinger Posts: 330 Registered: 4/30/07
Re: Prime numbers and pi
Posted: Apr 16, 2009 8:33 AM

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.