210 History of the Theory of Numbers. [Chap, vii 
Hoen4 Wronski^^^ stated without proof that, if a;'"=a (mod M), 
a = {-iy+'\hK+{-lY+''rA[M/K,o)Y-^+Mi, 
x = h + (-lY+'A[M/K, iry-'+Mj, 
and that M must be a factor of aK"" - \hK-{-iy+^\'". Here the "alephs" 
A[M/K, o)Y, for r = 0, 1,. . ., are the numerators of the reduced fractions 
obtained in the development of M/K as a continued fraction. In place of 
K, Wronski wrote the square of l''^^ = k\. Concerning these formulas, see 
Hanegraeff,"^ Bukaty,!^^ Dickstein.^^^ Cf. Wronski^^^ of Ch. VIII. 
E. Desmarest" noted that, if x'^+D=0 (mod p) is solvable, x^+Dy^ = mp 
can be satisfied by a value of m<3+y>/16 and a value of 2/^3. His proof 
is not satisfactory. 
D. A. da Silva''^ (Alasia, p. 31) noted that x^^l (mod m), where 
m = Pi*'P2^* •• ■ . has the roots 'Zxiqi7n/p{^ where Xi is a root of x^'= 1 
(mod Pi"^), Di being the g. c. d. of D and <t){pr), while the g's are integers 
such that Xqim/p['=\ (mod m). 
Da Silva^^^" proved that a solvable congruence a:"=r (mod m) can be 
reduced to the case r prime to m and then to the case m = p'',p a prime > 2. 
Then, if 5 is the g. c. d. of n and (f){p'')=8di, there is a root if and only if 
r*'=l (mod p") and hence if and only if r'^=l (mod p"'"^^), where p"' is the 
g. c. d. of n and p"~"\ while d is the quotient of p — 1 by its g. c. d. with n. 
H. J. S. Smith^'^" indicated a simplification in Gauss'^" second method 
of solving x^^A, If r^-\- D= (mod P) is solvable, mP = x^-\-Dy'^ is solvable 
for some value of ?72< 2V-D/3. Employing all values of m under that limit 
for which also 
(i)=S> 
and finding with Gauss all prime representations of the resulting products 
by the form x^-\-Dy^, we get ±r=x'/y', x"/y",. . .(mod P), where x', y'; 
x", y" \. . . denote the sets of solutions of mP = x^-\-Dy^. 
Eg. Hanegraeff^^^ reduced x"'=r to d"'r=l (mod p) by use of 6x=l. 
When p/d is developed into a continued fraction, let /x and P^_i be the 
number of quotients and number of convergents preceding the last. Let v, 
P^_i be the corresponding numbers for p/O"". Then 
x^i-iy-'P,_„ r=(-ir^P,_i (mod p). 
For p a prime, we get all roots by taking 6 = 1,. . . , (p — 1)/2. By starting 
with d{x — h)^l in place of 6x= 1, we get 
"'R6forme des Math^matiques, being Vol. i of R6forme du savoir humain, 1847. Wronski's 
mathematical discoveries have been discussed by S. Dickstein, Bibliotheca Math., (2), 
6, 1892, 48-52, 85-90; 7, 1893, 9-14 [on analysis, (2), 8, 1894, 49, 85; (2), 10, 1896, 5]. 
Bull. Int. Ac. Sc. Cracovie, 1896; Rozprawy, Krakow, 4, 1913, 73, 396. Cf. I'intermd- 
diaire des math., 22, 1915, 68; 23, 1916, 113, 164-7, 181-3, 199, 231-4; 25, 1918, 55-7. 
"'<KU. Alasia, Annaes Sc. Acad. Polyt. do Porto, 9, 1914, 65-95. There are many confusing 
misprints; for example, five at the top of p. 76. 
""British Assoc. Report, 1860, 120-, §68; CoU. M. Papers, 1, 148-9. 
"*Note BUT r^quation de congruence x^=r (mod p), Paris, 1860. 
