Meyer's theorem
In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation has a non-zero real solution, then it has a non-zero rational solution. By clearing the denominators, an integral solution may also be found.
Meyer's theorem is usually deduced from the Hasse–Minkowski theorem and the following statement:
Meyer's theorem is the best possible with respect to the number of variables: there are indefinite rational quadratic forms in four variables which do not represent zero. One family of examples is given by
where ' is a prime number that is congruent to 3 modulo 4. This can be proved by the method of infinite descent using the fact that, if the sum of two perfect squares is divisible by such a ', then each summand is divisible by .