210 HlSTOKY OF THE THEOKY OF NUMBERS. [CHAP. VII Hoen6 Wronski169 stated without proof that, if xm=a (mod M), and that M must be a factor of aKm - \ hK - ( - 1) k+l f m. Here the "alephs' ' A[M/K, co]r, 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 lfc/1 = A;!. Concerning these formulas, see Hanegraeff,171 Bukaty,180 Dickstein.194 Cf. Wronski151 of Ch. VIII. E. Desmarest37 noted that, if z2+D= 0 (mod p) is solvable, x2+Dy2 = mp can be satisfied by a value of w<3-f p/16 and a value of y^B. His proof is not satisfactory. D. A. da Silva42 (Alasia, p. 31) noted that xD=l (mod m), where ra=pi€lp2C2. • ., has the roots Ex^m/pi* where xt is a root of xDi^1 (mod piei), D{ being the g. c. d. of D and <t>(p/'i'), while the q's are integers such that S&m/p/^ 1 (mod ra) . Da Silva169a proved that a solvable congruence xn=r (mod m) can be reduced to the case r prime to m and then to the case m — pa)p a prime > 2. Then, if 8 is the g. c. d. of n and <Kpa) =^1? there is a root if and only if r5l==l (mod pa) and hence if and only if rd=l (mod p"'*1), where pa/ is the g". c. d. of n and pa~l, while d is the quotient of p — 1 by its g. c. d. with n. H. J. S. Smith170 indicated a simplification in Gauss'157 second method of solving £2E= A. If r2+D=0 (mod P) is solvable, mP = x2+Dy2 is solvable for some value of m< 2VD/3. Employing all values of m under that limit for which also and finding with Gauss all prune representations of the resulting products by the form x2+Dy2, we get =±r=x'/yr, xf'/y",.. .(mod P), where x', ytm, x", y";... denote the sets of solutions of mP = x2-{-Dy2. • Eg. Hanegraeff171 reduced xm^r to 0mr=l (mod p) by use of 6x^1. When p/6 is developed into a continued fraction, let /z and PM_! be the number of quotients and number of convergents preceding the last. Let v, Py_i be the corresponding numbers for p/6m. Then x^(-iy-lPM_1} rst-D'-ip^ (mod p). For p a prime, we get all roots by taking 6 = 1,..., (p —1)/2. By starting with Q(x—h)=I in place of 0x=l, we get 169R£forme des Mathematiques, being Vol. I of R6forme du savoir humain, 1847. Wronski's mathematical discoveries have been discussed by S. Dickstein, Bibliothcca Math., (2), 6, 1892,48-52,85-90; 7, 1893, 9-14 [on analysis, (2), 8, 1894, 49, 85; (2), 10, 1890, 5]. Bull. Int. Ac. Sc. Cracovie, 1896; Rozprawy, Krakow, 4, 1913, 73, 396. Cf. 1'intorrntf-diaire des math., 22, 1915, 68; 23, 1916, 113, 164-7, 181-3, 199, 231-4; 25, 1918, 55-7. 16SaC. 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. 170Britiah Assoc. Report, 1860, 120-, §68; Coll. M. Papers, 1, 148-9. 17INote BUT liquation de congruence xm=r (mod p), Paris, 1860.