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Topic: stairway numbers (revised version)
Replies: 6   Last Post: May 22, 2011 3:25 AM

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Franz Gnaedinger

Posts: 330
Registered: 4/30/07
Re: stairway numbers (revised version)
Posted: May 19, 2011 3:39 AM
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Euler summed the squares of the terms
of the harmonic series:

1 '4 '9 '16 '25 '36 '49 ... = '6 of pi x pi

He might have used the following stairway
(read 'n as 1/n and "n as 2/n) :

"1
'1x1 "5
'1x1 '2x2 "9
'1x1 '2x2 '3x3 "7
'1x1 '2x2 '3x3 '4x4 "9

A similar stairway approximates the natural
logarithm of 2 :

'1
'1x2 '5
'1x2 '3x4 '9
'1x2 '3x4 '5x6 '13
'1x2 '3x4 '5x6 '7x8 '17

1 minus '3
1 minus '2x3 '7
1 minus '2x3 '4x5 '11
1 minus '2x3 '4x5 '6x7 '15
1 minus '2x3 '4x5 '6x7 '8x9 '19

Where is the pattern that plays a role
for the primes? Is there such a pattern?
of a similar or a different kind?



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