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27 de julio de 2024
On the basic character of residue classes.
| Título inglés | On the basic character of residue classes. |
|---|---|
| Título español | Sobre el carácter básico de las clases de residuo. |
| Autor/es | Hilton, Peter J. ; Hooper, Jennifer ; Pedersen, Jean |
| Organización | Dep. Math. Sci. State Univ. New York, Binghamton (New York), Estados Unidos; Dep. Math. Univ. Utah, Salt Lake City (Utah), Estados Unidos;Dep. Math. Univ. Santa Clara California, Santa Clara (California), Estados Unidos |
| Revista | 0214-1493 |
| Publicación | 1989, 33 (2): 213-225, 6 Ref. |
| Tipo de documento | articulo |
| Idioma | Inglés |
| Resumen inglés | Let t, b be mutually prime positive integers. We say that the residue class t mod b is basic if there 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 non-residue 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. The t is basic mod b. We also give an algorithm for enumerating the (prime) numbers p lying in a give residue class mod 4t and satisfying d(p) = d. In an appendix we briefly discuss the case when b is even. |
| Clasificación UNESCO | 120501 |
| Palabras clave español | Teoría algebraica de números |
| Código MathReviews | MR1030965 |
| Código Z-Math | Zbl 0699.10009 |
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