254 History of the Theory of Numbers. [Chap, viii 
If 4r^ — 27s^ is a quadratic non-residue of p, the congruence has one and 
only one root; but if it is a residue, there are three roots or no root. 
G. Cordone^^^ gave simpler proofs of Oltramare's^^^ theorem II on the 
case fjL = Qn — I, gave theorems to replace VII and VIII, and proved that the 
condition in IX is sufficient as well as necessary. 
Ivar Damm^^ found when Cardan's formula gives three real roots, one 
or no real root of a cubic congruence, and expressed the roots by use of 
his quasi sine and cosine functions. For the prime modulus p = 3nH-l, 
f=x^-\-ax-\-b is irreducible if 
c = 
^isrea.,(-|+cf.*l. 
If p = 3n — 1, it is irreducible if c and ( — 6/2+0)" are both imaginary. 
There are given (p. 52) explicit expressions for h such that / is irreducible. 
J. Iwanow^^ gave another proof of the theorem of Woronoj.^^^ 
Woronoj ^^^ gave another proof of the same theorem and stated that the 
congruence has the same number of roots for all primes representable by 
a binary quadratic form whose determinant equals — 4r^+27s^. 
G. Arnoux^^*^ gave double-entry tables of the roots of the congruences 
x''+6x*+a=0 (mod m), and solved numerical cubic congruences by in- 
terpreting Cardan's formulas. 
G. Arnoux^^^ treated x^+6x+a=0 (mod m) by use of Cardan's formula. 
For m = 1 1 , he gave a table of the real roots for a ^ 10, 6^ 10, and the residues 
of 
^-4+27 
When R is a quadratic residue, the cube roots of — a/2=t y/R are found by 
use of a table for the Galois field of order 11^ defined by r=2 (mod 11), 
and the cubic is seen to have a real and two imaginary roots involving i. 
If jR is a quadratic non-residue, there are three real roots or none. Like 
results are said to hold when m — 1 is not divisible by 3. If w= 1 (mod 3), 
there is a single real root if 7? is a quadratic non-residue ; three real or three 
imaginary roots of the third order if ^ is a residue. 
L. E. Dickson'^^ proved that, if p is a prime >3, x^-\-^x-\-b=0 (mod p) 
has no integral root if and only if —4/3^ — 276^ is a quadratic residue of p^ 
say = 81)u^ , and if §( — 6+/xV — 3) is not congruent to the cube of any 
y+zy/ — 3, where y and z are integers. The reducible and irreducible 
cubic congruences are given explicitly. Necessary and sufficient conditions 
for the irreducibility of a quartic congruence are proved. 
'"Rendiconti Circolo Mat. di Palermo, 9, 1895, 221-36. 
"^BuU. Ac. Sc. St. Petersburg, 5, 1896, 137-142 (in Russian). 
'^Natural Sc. (Russian), 10, 1898, 329; of. Fortschritte Math., 29, 1898, 156. 
'3« Assoc, franc;, av. sc, 30, 1901, II, 31-50, 51-73; corrections, 31, 1902, II, 202. 
'"Assoc, frang. av. sc, 33, 1904, 199-230 [182-199], and Amoux'", 166-202. 
'"BuU. Amer. Math. Soc, 13, 1906, 1-8. 
