| Date: |
1989 |
| Abstract: |
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 tn ≡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. |
| Rights: |
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| Language: |
Anglès |
| Document: |
Article ; recerca ; Versió publicada |
| Published in: |
Publicacions matemàtiques, V. 33 n. 2 (1989) p. 213-225, ISSN 2014-4350 |
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