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




Re: stairway numbers (revised version)
Posted:
May 19, 2011 3:39 AM


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?



