212 History of the Theory of Numbers. [Chap, vii 
Ladrasch^^^ obtained known results on x^=a for any modulus. 
V. Bouniakowsky^"^ gave a method of sohdng q-S'^ ^r (mod P), where 
P is odd. His first illustration is 3'==±=1 (mod 25). Write the integers 
^(25 — 1)/2 in a line. Under the first four wTite in order the integers 
= (mod 3) ; under the next four write in reverse order those = 1 ; under 
the last four write in order those = 2. 
1* 2* 3* 4* 
3 6 9 12 
5 6* 7* 8* 
10 7 4 1 
9* 10 11* 12* 
2 5 8 11 
Mark with an asterisk 1 in the first line; below it lies 3; mark with an 
asterisk 3 in the first line; etc. The number 10 of the integers marked 
with an asterisk is the least solution x of 3""= —1 (mod 25). The sign is 
determined by the number of integers in the second set marked by an 
asterisk. The method applies to any P = 6n+1. But for P = 6n+5, we 
use for the second set of numbers in the second line those =2 (mod 3) in 
reverse order, and for the third set those =1 in order. If P = 23, we see 
that each of the 11 numbers in the first line are marked with an asterisk, 
whence 3^^=-l (mod 23). A like marking occurs for P = 5, 11, 17, 29. 
For P = 35, 12 numbers are marked, whence 12 is the least x for which 
3''=1 (mod 35). Starting with the unmarked number 5, we get the cycle 
5, 15, 10, whence 3^= —1 (mod 7); similarly, the cycle 7, 14 gives 3"= —1 
(mod 5). 
For g'-3'^=='=4 (mod 25), we begin with 4 in the second row. Since it 
hes below 7, we mark 7 with an asterisk in the second row; etc. We use an 
affix n on the number which is the nth marked by an asterisk. 
12 3 4 
5 
6 7 8 
9 10 
11 12 
3*6 g*3 9*5 12*10 
10 
7*2^*1 ]^*7 
2*^ 5 
8*8 11*9 
For 5 = 11, we have the entry 8*^ below 11; hence 11-3^=— 4, the sign 
following from the number of entries ^ 8 in the second set which are marked 
with an asterisk. Similarly for any 5^ 12, except g = 5, 10. 
Bukaty^^" discussed the formula of Wronski.^^^ 
T. N. Thiele^^^ used a mosaic (empty and filled squares on cross-section 
paper) to test y^=d{T[\od c), where c is an integer or Gauss complex integer 
a + 5v— 1, employing the graph oi y'^ — cx = d. 
Dittmar^^^ discussed a;^=r (mod p). Using Cauchy's^* explicit con- 
gruence for the numbers belonging to a given exponent, he gave the expanded 
form of the congruence with the roots belonging to the successive exponents 
1,. ..,21. 
"*Von den Kubischen Resten u. Nichtresten, Progr., Dortmund, 1870. 
i-'Bull. Ac. Sc. St. Pdterebourg, 14, 1870, 356-375. 
i*'>D6duction et demonstration de trois lois primordiales de la congruence des nombres, Paris, 
1873. 
*""0m Talmonstre," Forhandl. Skandinaviske Naturforskeres, Kjobenhavn, 11, 1873, 192-5. 
"^Die Theorie der Reste, insbesondere derer vom 3. Grade, nebst einer Tafel der Kubischen 
Reste aller Primzahlen der Form 6m + 1 zwischen den Grenzen 1 und 100. Progr. Koln 
Gym., Berlin, 1873. 
