On the basic character of residue classes

• Autores: Peter Hilton, J. Hooper, J. Pedersen
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 33, Nº 2, 1989, págs. 213-225
• Idioma: inglés
• Títulos paralelos:
  • Sobre el carácter básico de las clases de residuo
• Enlaces
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• Resumen

  • 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.