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A General Constant Recognition Procedure

8. A General Constant Recognition Procedure In all of the cases mentioned above, the authors of the respective studies had ``hunches'' beforehand as to what form the resulting formulas might take. Frankly, some insight of sort is invaluable in avoiding what otherwise is an exponential explosion in the number of possible terms. It simply is not possible to perform integer relation searches with every conceivable term. In fact, if the constant is known to only limited precision, the number of terms that can be considered in an integer relation search may be limited to a handful.

Nonetheless, it does appear feasible to define procedures that are successful in recovering the analytic form of many constants that naturally appear in mathematical calculations. The authors present the following procedure as an example:

Using PSLQ and full precision, check if is algebraic of degree n, for j up to m.
Using PSLQ and full precision, check if is given by a multiplicative formula of the form

where is the k-th prime, and where are a selected set of transcendentals.
Using PSLQ and quad precision, check if is given by a linear formula of the form

where and are each a product of up to three constants from a set of algebraic and/or transcendental constants.
If a tentative relation is found in the previous step using quad precision, then check it using full precision.

Some examples of constants recognized by above procedure are the following:

The root near 1.3851367 of the polynomial
The definite integral
The definite intregral
The definite integral