Chap. VIII] CUBIC CONGRUENCES. 255 
D. Mirimanoff^^^ noted that the results by Arnoux"^'^" may be com- 
bined by use of the discriminant D= —4b^ — 27a^= —3-Q^R in place of R, 
since — 3 is a quadratic residue of a prime p = Sk-{-l, non-residue ofp = Sk — l, 
and we obtain the result as stated by Voronoi.-^^^ 
To find which of the values 1 or 3 is taken by v when D is a quadratic 
residue, apply the theorem that if /(x) = (mod p) is an irreducible con- 
gruence of degree n and if Xq is one of its imaginary roots (say one of the 
roots of the equation f{x) = 0) , the roots are 
p pn—l 
Hence a function unaltered by the cycHc substitution (xqXi. . .Xn-i) has 
an integral value modulo p. Take w = 3, D=d^, a a root 5^1 of 2^=1 
(mod p), and let 
M = (xo+aXi+a^X2)^. 
If p=l (mod 3), a is an integer, and M is an integer if ^^ = 1, while M is 
the cube of an integer if v = S. Thus we have Arnoux's criterion :^^^ v = S 
if ilf or-| ( — 9a+V — 3d) is a cubic residue modulo p. If p= —1 (mod 3) 
J/ = 3 if and only if ilf^^l (mod p), where k = {p'^ -l)/3. 
For quartic congruences, we can use (a^o — a:i-|-a:2 — a^s)^. 
R. D. von Sterneck^"*" noted that if p is a prime >3 not dividing A, 
and if k = SAC — B^^O (mod p), then the number of incongruent values 
taken by Ax^+Bx^+Cx-\-D is i{2p+(-3/p)j ; but, if k=0, the number 
is p if p = Sn — l, (p+2)/3 if p = 3n+l. Generalization by Kantor.^^^ 
C. Cailler^^^ treated x^+px-\-q=0 (mod I), where I is a prime >3. By 
the algebraic method leading to Cardan's formula, we write the congruence 
in the form 
(1) x^-Sabx+abia+b)^0 (mod /), 
where a, h are the roots of z^-{-Sqz/p—p/S=0 (mod T), whence 
z={xQ-\-aXi+ a^X2) V (9p) , a^ + a + 1 = (mod I) . 
Let A = 4p^+27g^. If 3A is a quadratic residue of i, a and b are distinct 
and real. If 3A is a non-residue, a and b are Galois imaginaries r±s\/iV^ 
where N is any non-residue. For a root a: of (1), 
Use is made of a recurring series S with the scale of relation [a +6, —ab] 
to get 2/0, Vi,.. .. Write Q = (3A/Z). If ; = 3m-l, Q = l, then 
If l = 3m-\-l, Q = l, the congruence is possible only when the real number 
a/b is a cubic residue, i. e., if 2/^=0 in S; let a/b belong to the exponent 
3m=f1 modulo I, whence 
'"L'enseignement math., 9, 1907, 381-4. 
"oSitzungsber. Ak. Wiss. Wien (Math.), 116, 1907, Ila, 895-904. 
"'L'enseignement math., 10, 1908, 474r-487. 
