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.
@article{urn:eudml:doc:41096,
title = {On the basic character of residue classes.},
journal = {Publicacions Matem\`atiques},
volume = {33},
year = {1989},
pages = {213-225},
zbl = {0699.10009},
mrnumber = {MR1030965},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41096}
}
Hilton, Peter J.; Hooper, Jennifer; Pedersen, Jean. On the basic character of residue classes.. Publicacions Matemàtiques, Tome 33 (1989) pp. 213-225. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41096/