Bailey–Borwein–Plouffe formula

In mathematics, the Bailey - Borwein - Plouffe formula called ( BBP formula ) is a Canadian mathematician Simon Plouffe from 1995 discovered empirical formula for calculating the number of loop

The discovered Plouffe series for is:

The formula is named the magazine article by the authors David H. Bailey, Peter Borwein, and Simon Plouffe, in which the formula was first published. The amazing thing about this particular formula is that one can derive an algorithm from it with a little rearranging, the measures any digit of the representation of the hexadecimal system without calculating the previous digits ( digit - extraction).

Polylogarithmische constant

Since Plouffes discovery of many similar formulas of the form were

Discovered which add up to other fundamental mathematical constants (as shown to the base ), such as the constants and polylogarithmic Catalanschen constant G. formulas and is referred to as BBP series to the base b. The question to which mathematical constants BBP rows exist, is still unresolved. For prime numbers p exists for a BBP series:

23, 47, 53 and 59 are the smallest prime numbers that are missing in this list. However, it is unproven whether to actually exists no BBP series. Presumably there are no BBP series for square roots, the Euler number e and the Euler constant, as this ( presumably) are not polylogarithmic constants.

BBP algorithm

An example is to show how to get the digits of a number representation. Thus one gets, for example, the fourth decimal place of by

• Multiplication with ...

• Cutting away the integer part ...

• Multiplication with ...

• And cutting away the broken part ...

Analogously, the digits of the hexadecimal representation of to

Multiplying the Plouffe formula gives with after division into four terms

Since the expression is received for only the fractional part of, we first remove from each summand the integer part. In the first summand of we achieve this by applying the operator to the counter, the remaining summands with have no integer part. In order to change from to

And it is

Using the symbol for congruence. Since the modified sums itself and certainly their linear combination may contain integers, they must be removed. Then you can multiply and cut away the broken part to get eventually.

Advantages of the BBP algorithm

This method to extract only the location of currently required, saves space for the previous points. Further, one can use simpler data types for the storage of the acquired points, in turn, also have shorter access times, which makes the algorithm faster, ultimately. Therefore, this method has made all previous algorithms for computing (the larger and more complex data types needed ) superfluous.