# On the basic character of residue classes.

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

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## Abstract

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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.

## How to cite

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Hilton, 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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