Let t, b be mutually prime positive integers . We say that the residue class t mod b is basic if theie exists n such that ta --1 mod b; otherwise t is not basic. In this paper we relate the basic character of t mod b to the quadratic character of t modulo the prime factors of b. If all prime factors p of b satisfy p -3 mod 4, then t is basic mod b if t is a quadratic nonresidue mod p for all such p ; and t is not basic mod b if t is a quadratic residue mod p for all such p. If, for all prime factors p of b, p -1 mod 4 and t is a quadratic non-residue mod p, the situation is more complicated . We define d(p) to be the highest power of 2 dividing (p -1) and postulate that d(p) takes the same value for all prime factors p of b. Then t is basic mod b. We also give an algorithm for enumerating the (prime) numbers p lying in a given residue class mod 4t and satisfying d(p) = d. In an appendix we briefly discuss the case when b is even . IntroductionIn a series of papers [1,through 4], culminating in the monograph [5], Hilton and Pedersen developed an algorithm -in fact, two algorithms, one being the reverse of the other -for calculating the quasi-order of t mod b, where t, b are mutually prime positive integers, and determining whether t is basic mod b . Here the quasi-orden óf t mod b is the smallest positive integer k such that t k -± 1 mod b ; and t is said to be basic if, in fact, tk --1 mod b . Thus t is basic if and only if the order of t mod b is twice the quasi-order of t mod b (in the contrary case the quasi-order and the order coincide. Froemke and Grossman carried the number-theoretical investigation considerably further in [6] and drew attention to the importance, where b is prime, of the quadratic character of t mod b in their arguments .Our object in this paper is to relate the basic character of t mod b to the quadratic character of t modulo the prime factors of b. We assume b odd, but add a few remarks in an appendix on the case when b is even .Given a pair (t, p) where p is an odd prime not dividing t, we distinguish 4 possibilities as follows : 9t may or may not be a quadratic residue mod p, and we may have p -1 mod 4 or p -3 mod 4. We restrict attention, in our
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