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New Formulas for Pi and Related Constants

7. New Formulas for Pi and Related Constants Many readers may be already familiar with the recent paper by the authors and others which gives a new method for computing individual hexadecimal digits of

, as well as a number of other constants. A brief review of these results is as follows. First, let us ask whether

satisfy a relation of the form

where

are rational numbers? Indeed it does. Such a formula can be found by separating the right hand side of the above expression into eight summations, numerically evaluating these to high precision, appending the numerical value of

, and applying PSLQ to the resulting 9-long vector. When this is done, PSLQ discovers the following formula:

A similar formula was discovered by Ferguson shortly after the discovery of the above formula. In fact, there is a two-dimensional lattice of such formulas, which lattice can be generated by these two formulas.

The significance of these formulas for the computation of can be seen as follows. Let be the first of the sums in the above formula for . Then we can write

The first sum can be rapidly evaluated by means of the binary algorithm for exponentiation, where each operation is performed modulo the integer 8k + 1. These calculations can be done with either integer or floating-point arithmetic, provided the format being used has enough accuracy to exactly represent the integer

. Once an individual exponentiation operation is complete, the resulting integer value is divided by 8 k + 1, using floating-point arithmetic, and added to the sum modulo 1. Only a few terms are required of the second, since the terms rapidly become smaller than the ``machine epsilon'' of the floating-point arithmetic system being used. The resulting fractional value, when expressed in base 16 notation, gives the hexadecimal digits of

beginning at position d + 1. Here are a number of other formulas of this type. As before, these formulas were originally found using PSLQ searches.

Full details of these calculations, as well as formal proofs of the above formulas, can be found in [5].

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