Quadratic and Linear Congruence*

The number of simultaneous solutions of a quadratic and a linear congruence does not seem to be discussed in the literature, yet a knowledge of the invariants necessary to specify this number should lead to an arithmetical classification of the form-pairs involved. This preliminary investigation is confined to congruences with modulus odd and prime to the g.c.d.'s of the two sets of coefficients. From the formulas obtained, a simple use of the Chinese Remainder Theorem will give the number of solutions for any such modulus which is either square-free or at least whose prime factors of power greater than the first are of a definite class. An interesting application, of a different type from the preceding, is given in §6. Special cases of this and of Theorem 1 have already been proven.f