Integer Relations

Let $a\in\mathbb R^n$ be a given vector. We say that the vector $c\in\mathbb Z^n$ is an integer relation for a if

$$c_1a_1+c_2a_2+\cdots+c_na_n=0$$ (1)

with at least one c_i different from zero. It is the purpose of integer relation algorithms to search for a non-zero vector csatisfying (1), possibly with a certain level of confidence.

Of course, a numerical discovery of a relation by a computer does not in general constitute a proof of this relation; one of the reasons being that the computer operates on rational approximations of numbers that in many applications are likely to be transcendental. In many cases, however, the relations we first discovered numerically subsequently received rigorous mathematical proofs: many complicated relations probably would never have been dreamed of without the assistance of the computer. In Section 3 we show examples of such phenomena.