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Fibonacci Formula for Pi

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Fibonacci Formula for Pi

Here’s an unusual formula for pi based on the product and least common multiple of the first m Fibonacci numbers.

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Here’s an unusual formula for pi based on the product and least common multiple of the first m Fibonacci numbers. 

\pi = \lim_{m\to\infty} \sqrt{\frac{6 \log F_1 \cdots F_m}{\log \mbox{lcm}( F_1, \ldots, F_m )}}

Unlike the formula I wrote about a few days ago relating Fibonacci numbers and pi, this one is not as simple to prove. The numerator inside the root is easy enough to estimate asymptotically, but estimating the denominator depends on the distribution of primes.

Source: Yuri V. Matiyasevich and Richard K. Guy, A new formula for π, American Mathematical Monthly, Vol 93, No. 8 (October 1986), pp. 631-635.

Topics:
fibonacci ,pi ,formulae ,math ,data

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