If [DELTA] is quadratic nonresidue then [alpha](p) | p + 1.
If [DELTA]] is a quadratic nonresidue modp, then [mathematical expression not reproducible].
A is a quadratic nonresidue mod p; then [[beta].sup.p+1] = [[alpha].sup.p+1].
Otherwise a is quadratic nonresidue
modulo n and denoted as a [member of] [bar.[Q.sub.n]]
This phenomenon may be explained by the fact that, for symmetric matrices with corresponding quadratic residue [DELTA], there are symmetric matrices with corresponding nonresidue
A' (Lemma 1).
The additional production that is forecast for the pellet industry globally may well involve roundwood (nonresidue
) raw materials.
Hereafter referred to as nonresidue, Flemingia leaf, medic hay, and wheat straw, respectively.
At the 48-day harvest, application of medic hay, Flemingia leaf, and wheat straw in the presence of N alone produced increased plant top yield compared with the nonresidue control.
Therefore (24[Q.sub.n] + 1/179) = (128/179), (105/179) or (53/179), showing (24[Q.sub.n] + 1/179) = -1, since 128, 105 and 53 are quadratic nonresidues modulo 179, so that we can eliminate n [equivalent to] 15, 25 and 51 (mod 72).
If n [equivalent to] 24, 39, 48 or 49 (mod 72), then 24[Q.sub.n] + 1 [equivalent to] 24[Q.sub.n] + 1, 24[Q.sub.39] + 1, 24[Q.sub.48] + 1 or 24[Q.sub.49] + 1 [equivalent to] 62, 52, 62 or 31 (mod 73), by Table 3(a) and all of them are quadratic nonresidues modulo 73.
Note that for every nonzero element a [element of] [Z.sub.p], the set aS (mod p) is either the set of all quadratic residues or the set of all quadratic nonresidues modulo p.
Lower bounds for least quadratic nonresidues. In Analytic Number Theory: Proceedings of a Conference in Honor of P.