** Next:** Primes suitable for nested
** Up:** A better method for
** Previous:** A better method for

It is important to check that these numbers, like *S*(*e*) above, do not
satisfy simple algebraic relationships.
It would be very unfortunate that we randomly select an *S*(*e*) which
happens to be, for example, *S*(*e*)=1/3.
This can be checked quite easily using the LLL lattice reduction
algorithm.
To do this we construct a matrix of dimension
with the
following structure:

The last row in the matrix is used to reduce the values by integer
multiples of *n*, as should be for (mod *n*) computations.
Every vector which is a linear combination of some of the
rows, will give a polynomial in *S*(*e*) of degree at most *m*-1 (mod *n*)
that is satisfied.
A *short* vector will give a simple polynomial, hence a suspicious
relationship which may be better avoided.
For the above choices of *S*(*e*) and *m*=5, the reduced lattice is

So the simplest polynomial relationship of degree four or less
satisfied by *S*(*e*) (from the last row) is
.
Under this scheme, we can now compute the trigonometric functions
with the standard complex-exponential formulas:

All the trigonometric relationships, including the relations with
the complex exponentials will be verified.
For the numerical example,
*S*_{n}(*i*)=917163,
*S*_{n2}(*i*)=649529 and *T*=2888692.

** Next:** Primes suitable for nested
** Up:** A better method for
** Previous:** A better method for
*Gaston Gonnet*

*1999-07-04*