256 History of the Theory of Numbers. [Chap, viii 
w= I 7 I J x= or > 
\o/ 2/2^-1 2/m 
according as the upper or lower sign holds. If l==Sm-\-l, Q= —1, then 
2/3m+2=0, (rl =-, realx=-; . 
\0/ y2m+l 
li l = Sm — l, Q= —1, there are three real roots if and only if a/b is a cubic 
residue of I, viz., 2/^=0; when real, the roots may be found as in the second 
case. 
Cailler^^^ noted that a cubic equation X = has its roots expressible 
rationally in one root and VA, where A is the discriminant (Serret's 
Algebre, ed. 5, vol. 2, 466-8). Hence, if p is a prime, X=0 (mod p) has 
three real roots if one, when and only when A is a quadratic residue of p. 
If p = 9m^l, his^^^ test shows that x^ — 3xH-l = (mod p) has three real 
roots, but no real root for other prime moduli 5^3. The function 
F{x) =x^-{-x- — 2x — l for the three periods of the seventh roots of unity is 
divisible by the primes 7m=*= 1 (then 3 real roots, Gauss®", p. 624) and 7, 
but by no other primes. 
E. B. Escott"^ noted that the equation F(x) =0 last mentioned has the 
roots a, ^ = a^—2, 7=/3^— 2, so that F{x) = (mod p) has three real roots 
if one real root. To find the most general irreducible cubic equation with 
roots a, (3, y such that 
^=/(a), y=m, a=f{y), 
we may assume that/(x) is of degree 2. For /(a) = a^—n, we get 
(2) x^-\-ax'^-{a'-2a-\-3)x-ia^-2a^-\-da-l)=0, 
with ^ = a^ — c, y=j3^—c, a=y^—c, c = o^ — a+2. The corresponding con- 
gruence has three real roots if one. To treat/(a)=a^+^a+Z, add k/2 to 
each root. For the new roots, jS' = a'^ — n, as in the former case. To treat 
/(a) = ta^-\-ga-\-h, the products of the roots by t satisfy the preceding relation. 
L. E. Dickson^^ determined the values of a for which the congruence 
corresponding to (2) has three integral roots. Replace x by z—a; we get 
z^-2az^+{2a-S)z-\-l = (mod p). 
If one root is z, the others are 1 — 1/z and 1/(1—2). Evidently a is rational 
in z. If —3 is a quadratic non-residue of p, there are exactly (p — 2)/S 
values of a for which the congruence has three distinct integral roots. If 
— 3 is a residue, the number is (pH-2)/3. A second method, yielding an 
explicit congruence for these values of a, is a direct application of his^^® 
general criteria for the nature of the roots of a cubic congruence. 
T. Hayashi^^^ treated cyclotomic cubic equations with three real roots 
by use of Escott's^'*^ results. 
"«L'interm^diaire des math., 16, 1909, 185-7. '"/bid., (2), 12, 1910-11, 149-152. 
'"Annals of Math., (2), 11, 1909-10, 86-92. >«/6id., 189-192. 
