Does the summation of reciprocal prime pair products

 

1/(2*3)+1/(3*5)+1/(5*7)+1/(7*11)+1/(11*13)+1/(13*17)...=?

 

or the summation of reciprocal Fibonacci pair products

 

1/(1*1)+1/(1*2)+1/(2*3)+1/(3*5)+1/(5*8)+1/(8*13)+1/(13*21)...=?

 

have any significance in number theory?

 

:inf:

:lsum: F_n/F_n-1 = :lcphi:, the Golden Ratio

n=1

 

and another way a similar series equals :lcphi::

 

:inf:

:llb::lsum: ((-1)ⁿ⁺¹)/(F_nF_n+1):lrb:+1 = :lcphi:

n=1

and further more i worked out that

:inf:

:lsum:1/F_2nF_2n+2 = _:lcphi:^-2

n=1

This works out mathematicaly but is it true in the microcosom and macro? Math is perfect in our observation of our general existance but it seems something throws a monkey wrench into the works at the limits of our existance.

Thanks for the math guys.

Just aman

aman~

It might be more mathematicaly correct to say that our existence IS THE RESULT of the monkey wrench. The universe is all about symmetry. Of course, if everything WAS perfectly symetrical, there wouldn't be any you or me. Disturbences in the universal symmetry create chaos and pattern (patterns are an expression of the underlying symmetry). And you and I are just patterns-- I'm simplifying, of course. My point is that math isn't just "perfect in general but fuzzy 'round the edges". It actually does a nice job of explaining all that monkey-wrenchedness.