A set of real numbers , ..., is said to possess an integer relation if there exist integers
such that
with not all .
For historical reasons, integer relation algorithms are sometimes called generalized
Euclidean algorithms or multidimensional continued fraction algorithms.
An interesting example of such a relation is the 17-vector (1, ,
, ..., ) with , which has an integer relation (1, 0, 0, 0,
, 0, 0, 0, , 0, 0, 0, , 0, 0, 0, 1), i.e.,
This is a special case of finding the polynomial of degree satisfied by .
Integer relation algorithms can be used to solve subset sum problems, as well as to determine if a given numerical constant is equal
to a root of a univariate polynomial of degree or less (Bailey and Ferguson 1989, Ferguson and Bailey 1992).
One of the simplest cases of an integer relation between two numbers is the one inherent in the definition of the greatest common divisor.
The well-known Euclidean algorithm solves
this problem, as well as the more general problem of an integer relation between
two real numbers, yielding either an exact relation or an infinite sequence of approximate
relations (Ferguson et al. 1999). Although attempts were made to generalize
the algorithm to
by Hermite (1850), Jacobi (1868), Poincaré (1884), Perron (1907), Brun (1919,
1920, 1957), and Szekeres (1970), all such routines were known to fail in certain
cases (Ferguson and Forcade 1979, Forcade 1981, Hastad et al. 1989). The first
successful integer relation algorithm was developed by Ferguson and Forcade (1979)
(Ferguson and Bailey 1992, Ferguson et al. 1999).
Plouffe's "Inverse Symbolic Calculator" site includes a huge database of 54 million real numbers which are algebraically related
to fundamental mathematical constants. The Ferguson-Forcade
algorithm has shown that there are no algebraic equations of degree with integer coefficients having Euclidean norms below
certain bounds for ,
, , , , , , and , where e is the
base for the natural logarithm, is pi, and is the Euler-Mascheroni
constant (Bailey 1988).
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