On the Basic Character of Residue Classes

ON THE BASIC CHARACTER OF RESIDUE CLASSES P . HILTON, J . HOOPER AND J . PEDERSEN 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 . 0. Introduction In 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 214 P . HILTON, J . HOOPER, J . PEDERSEN discussion of the basic character of t mod b, to the situation in which all prime factors p of b place (t, p) in the same class . If t is a quadratic residue mod p and p 1 mod 4, we are unable to draw any general conclusion about the basic character of t mod p . Thus, for example, 77 1 mod 29, 77 -1 mod 113, and 7 is a quadratic residue modulo 29 or 113 . If all prime factors p of b satisfy p 3 mod 4, it is easy to draw general conclusions about the basic character of t mod b; our results are given in Section 2. The most interesting case for our purposes is that in which t is not a quadratic residue mod p and p 1 mod 4, for all prime factors p of b. It then becomes important to be able to calculate the function d(p), where d(p) = d if p = 1 +2de, with e odd . Thus d is a positive integer and, in fact, d >_ 2 in the case we are discussing . Since the quadratic character of t mod p depends only en the residue class of p mod 4t, we give an algorithm for enumerating those primes p, as functions of s and d, such that (0 .1) d(p) = d, p s mod 4t, 1 < s < 4t 3 . If the prime factors p of b are confined to those satisfying (0.1) for fixed s, d, then, as we show in Section 3, t is basic mod b. In Section 1 we announce some elementary results which are used in proving our main theorems . Throughout the papes we use the symbol e for a number which is +1 or -1 . 1 . Some preliminary lemmas The first result extends to the quasi-order a familiar result en order. Lemma 1 .1 . Let the quasi-order of t mod b be n and let te mod b. Then nim . Proof. Let m = qn + r, 0 <_ r < n, and tn 77 mod b, q = ±1. Then t , = tm (t , )-9 = 6779 = ±1 mod b, so that r = 0. We now restrict b by the condition b >_ 3, so that the basic character of t mod b comes into question . Lemma 1 .2 . The residue t mod b is basic if and only if t'n -1 mod b for some exponent m. Proof. The necessity of the condition is obvious . Suppose then that t'n = -1 mod b and that the quasi-order of t mod b is n. Then n 1 m, by Lemma 1 .1 . Thus, if tn = 1 mod b, it follows that t' = 1 mod b. This contradiction shows that tn -1 mod b, so that the residue t mod b is basic . THE BASIC CHARACTER oF RESIDUE CLASSES 215 Lemma 1 .3 . The residue t mod b is non-basic if t'n 1 mod b for some odd exponent m. Proof.. Let the quasi-order of t mod b be n with tn e mod b. Then n 1 m, so that m = nq. Since m is odd, q is odd . Thus t'n e9 = e mod b, so e = 1 and the residue t mod b is non-basic . Our next result is of a different kind . Proposition 1 .4 . Le¡ x y mod m. Then XMk-1 = ymk-1 mod mk , k>1. Proof. We argue by induction on k, the case k = 1 being trivial . If we assume xmk1 = ymk-1 mod mk for a certain k > 1, then xmk-1 = ymk-1 + Am k, so that mk mk-1 + Amk m = ymk + AMk+1ym k-1 (m-1) + r m l ~\2 m2kymk-1 (m-2) + 22 y m mod mk+1 . This establishes the inductive step, and hence the proposition . We have the immediate consequence : Lemma 1 .5 . Let c e mod m, with m odd. Then cmk-1 = e mod mk, k > 1 . Proof. We have only to note that e"k-1 = e if m is odd . 2. The main results We recall the following key results on quadratic reciprocity. Theorem 2.1 (Euler). Le¡ p be an odd prime. Then (i) t p 21 1 mod p if and only if t is a quadratic residue mod p (ii) t p2 1 -1 modp if and only if t is not a quadratic residue mod p. Theorem 2 .2 (Gauss) . Let p be an odd prime. Then the quadratic character of t mod p depends only on the residue class of p mod4t and is the lame for two odd primes p and q such that p -q mod4t . 216 P . HILTON, J . HOOPER, J . PEDERSEN Thus, given t and p, we distinguish 4 classes into which the pair (t, p) may fall : I p 1 mod 4, t P 2 l mod p; II p-1mod4,t _21 --1 mod p ; IIIp-3mod4,t 2 1modp; IV p 3 mod 4, t_21 T -1 mod p ; We will say nothing further about residues t mod b if b admits a factor p such that (t, p) is in class I . We will henceforth, until otherwise stated, assume that b is odd. Theorem 2.3 . Suppose that ¡he prime factors p of b are all such that (t, p) is in Class III. Then the residue t is not basic mod b . Proof. Let b = Hay=1Pk¡, k i > 1 . Then is odd and t P, 2 1 1 mod pi . By Lemma 1 .5, t1 1 Pk' -I 1 mod pi' . Set m = IIN1(zl )Pk` -1 . Then m is odd, and 1 mod pk` . It follows that t"° 1 mod b, so that, by Lemma 1.3, the residue t is not basic mod b. Theorem 2 .4 . Suppose that the prime factors p of b are all such that (t, P) is in Class IV Then the residue t is basic mod b. Proo£ We argue as for Theorem 2 .3, except that now t' -1 mod P;', t' -1 mod b, with m odd . We apply Lemma 1 .2 to obtain the result . Example 2.1 . Let t = 7 . Then, by Theorem 2.2, we must consider primes p mod 28 . We easily find p 1 or 27 mod 28 : 7 p 2 1 1 mod p p 3 or 25 mod 28 : 7 p 2 1 =1 mod p p 5 or 23 mod 28 : 7-21 -1 mod p p 9 or 19 mod 28 : 7 , z 1 1 mod p p 11 or 17 mod 28 : 7-= -1 mod p p 13 or 15 mod 28 :7-21 = -1 mod p THE BASIC CHARACTER OF RESIDUE CLASSES 217 Thus (7, p) is in Class I if p 1, 9, 25 mod 28 ; (7,p) is in Class II if p 5,13,17 mod 28 ; (7,p) is in Class III if p 3,19, 27 mod 28 ; . (7,p) is in Class IV if p 11, 15,23 mod 28 . We conclude that 7 is not basic mod b if b is a product of primes p such that p 3,19 or 27 mod 28 ; and 7 is basic mod b if b is a product of primes p such that p 11,15 or 23 mod 28 . As we have said, no inference can be drawn if b is a product of primes p such that p 1, 9 or 25 mod 28 . Indeed, the fact that (7, p) is then in Class I is a special case of the following phenomenon, which we describe here for the sake of completeness . Proposition 2 .5 . Le¡ p be an odd prime such that p = kz -}41 . Then any factor of l is a quadratic residue mod p. Proof. It suffices to prove this for prime factors q of l . Now if q = 2, then p 1 mod 8, so 2 is a quadratic residue mod p. If q is odd, then p is a quadratic residue mod q and p z l is even, so that, by quadratic reciprocity, q is a quadratic residue mod p . Note that it follows, by Theorem 2.1, that 72 1 mod p if p or 25 mod 28 . We will devote the next section to a discussion of the Class II . At this point, we are content to remark Theorem 2.6 . Suppose that b = residue t is basic mod b. Proof. t ( P21 )pk-1 -1 mod pk . Apply Lemma 1 .2 . In the next section we generalize this obvious conclusion . 3. The class II situation We define a function d from positive integers >_ 2 to non-negative integers by (3 .1) d(n) = d !--1 2d 1 (n 1), 2d+1X(n 1) . where (t, p) is in Class II. Then the Notice that, for an odd prime p, d(p) >_ 1 and that d(p) >_ 2 if (t, p) is in Class II . Let d be a fixed integer >_ 2 ; we then have the following theorem, generalizing Theorem 2.6 . 220 P . HILTON, J . HOOPER, J . PEDERSEN in this way, splitting the inequality d(p) > d into the two possibilities d(p) = d or d(p) > d + 1 . We demonstrate this tree in Figure 1 . mod 1+2N b, N>M 2M a d>M 1+2N b 1+2Nb+2Ma 2M+la if N =M stops, d = M goes on, d >_ M+ 1 if N > M goes on, d > M+ 1 stops, d = M General Case Figure 1 p 5 mod 28 Special Case Figure 1 The tree provides the conceptual basis for the proof (see Theorem 3.2),that the primes p satisfying the congruence p 2dc + 1 mod 2d+iu constitute the totality of the primes p satisfying p s mod 4t and d(p) = d. For we may argue by induction on d that there is only one residue class mod 2d+iu containing such primes . We assume n > m + 2 and we rewrite (3.4) as (3.11) 2dc(d) 2nv mod u, THE BASIC CHARACTER op RESIDUE CLASSES 221 to emphasize the dependence of c on d; recall d >_ m+2. We start the induction (and the tree) by rewriting p s mod 4t, using (3.11), as (3.12) p 1+2-+2C(m -}2) mod 2rn+2u . Then (3 .12) branches into the two congruences