The Probability That a Determinant Be Congruent

The problem treated is the evaluation of Pn(a, m)t the probability that a determinant of order n having integral elements be congruent to a modulo m. By "probability" is meant the fraction obtained by dividing the number of favorable cases by the number of possible cases: let each element of the determinant range over the values 1, 2, • • • , m; among the ra" possible determinants let g be the number which are congruent to a modulo m; then Pn(a, m) =g/m . This problem has been investigated by Jordan, whose solution involves the function