# On the basic character of residue classes.

Peter J. Hilton; Jennifer Hooper; Jean Pedersen

Publicacions Matemàtiques (1989)

- Volume: 33, Issue: 2, page 213-225
- ISSN: 0214-1493

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topHilton, Peter J., Hooper, Jennifer, and Pedersen, Jean. "On the basic character of residue classes.." Publicacions Matemàtiques 33.2 (1989): 213-225. <http://eudml.org/doc/41096>.

@article{Hilton1989,

abstract = {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.},

author = {Hilton, Peter J., Hooper, Jennifer, Pedersen, Jean},

journal = {Publicacions Matemàtiques},

keywords = {Teoría algebraica de números; basic character; quadratic nonresidue},

language = {eng},

number = {2},

pages = {213-225},

title = {On the basic character of residue classes.},

url = {http://eudml.org/doc/41096},

volume = {33},

year = {1989},

}

TY - JOUR

AU - Hilton, Peter J.

AU - Hooper, Jennifer

AU - Pedersen, Jean

TI - On the basic character of residue classes.

JO - Publicacions Matemàtiques

PY - 1989

VL - 33

IS - 2

SP - 213

EP - 225

AB - 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.

LA - eng

KW - Teoría algebraica de números; basic character; quadratic nonresidue

UR - http://eudml.org/doc/41096

ER -

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