On the complementary factor in a new congruence algorithm

In an earlier paper the authors described an algorithm for determining the quasi-order, Q t ( b ) , of t mod b , where t and b are mutually prime. Here Q t ( b ) is the smallest positive integer n such that t n = ± 1 mod b , and the algorithm determined the sign ( − 1 ) ϵ , ϵ = 0 , 1 , on the right of the congruence. In this sequel we determine the complementary factor F such that t n − ( − 1 ) ϵ = b F , using the algorithm rather that b itself. Thus the algorithm yields, from knowledge of b and t , a rectangular array a 1 a 2 … a r k 1 k 2 … k r ϵ 1 ϵ 2 … ϵ r q 1 q 2 … q r The second and third rows of this array determine Q t ( b ) and ϵ ; and the last 3 rows of the array determine F . If the first row of the array is multiplied by F , we obtain a canonical array, which also depends only on the last 3 rows of the given array; and we study its arithmetical properties.