212 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vii Ladrasch178 obtained known results on afea for any modulus. V. Bouniakowsky179 gave a method of solving g-3z= =«=r (mod P), where P is odd. His first illustration is 3*==*=! (mod 25). Write the integers ^(25 —1)/2 in a line. Under the first four write in order the integers =0 (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 3Z= — 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+l. But for P = 6n+5, we use for the second set of numbers hi the second line those =2 (mod 3) hi 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 3us=-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 3X=1 (mod 35). Starting with the unmarked number 5, we get the cycle 5,15, 10, whence 33= -1 (mod 7); similarly, the cycle 7,14 gives 32= -1 (mod 5). For g-3*==±=4 (mod 25), we begin with 4 hi the second row. Since it lies 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. 1234 3*6 6*3 9*5 12* 5678 10 7*24*11*7 9 10 11 12 2*4 5 g*8n*9 For g = ll, we have the entry 8*8 below 11; hence ll-38= — 4, the sign following from the number of entries ^ 8 in the second set which are marked with an asterisk. Similarly for any q^ 12, except g = 5, 10. Bukaty180 discussed the formula of Wronski.169 T. N. Thiele181 used a mosaic (empty and filled squares on cross-section paper) to test y2^d(mod c), where c is an integer or Gauss complex integer a-j-bV — 1, employing the graph of y2—cx = d. Dittmar182 discussed ofer (mod p). Using Cauchy's14 explicit congruence for the numbers belonging to a given exponent, he gave the expanded form of the congruence with the roots belonging to the successive exponents 178Von den Kubischen Resten u. Nichtresten, Progr., Dortmund, 1870. 179Bull. Ac. Sc. St. PStersbourg, 14, 1870, 356-375. 180D6duction et demonstration de trois lois primordiales de la congruence des nombres. Paris, 1873. 181(1 Om Talmonstre," Forhandl. Skandinaviske Naturforskeres, Kjobenhavn, 11, 1873, 192-5. 182Die Theorie der Reste, insbesondere derer vom 3. Grade, nebst einer Tafel der Kubischen Reste aller Primzahlen der Form 6n-f-l zwischen den Grenzen 1 und 100. Progr. K6ln Gym., Berlin, 1873.