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Full text of "History Of The Theory Of Numbers - I"

CHAP. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS.    275 A. Fleck103 proved that
if and only if p is a prime.   The case a= 1 is Wilson's theorem. Gu&in104 asked if Wolstenholme's73 result is new and added that
THE CONGRUENCE 1-2-3. . .(p—l)/2==*=l (MOD p).
J. L. Lagrange110 noted that p—1, p— 2,. . ., (p+l)/2 are congruent modulo p to —1, —2,..., — (p— 1)/2, respectively, so that Wilson's theorem gives
(4)                          (l-2.3. . .Eziy^-l)^ (mod p). For p a prune of the form 4n+3, he noted that
(5)                                    1-2-3... ^jps*! (modp).
E. Waring111 and an anonymous writer112 derived (4) in the same manner.
G. L. Dirichlet113 noted that, since —1 is a non-residue of p = 4n-f 3, the sign in (5) is + or — , according as the left member is a quadratic residue or non-residue of p. Hence if m is the number of quadratic non-residues <p/2ofp,
1-2-3... £=is(-ir (modp).
C. G. J. Jacobi114 observed that, for p>3, m is of the same parity as N, where 2N— 1 = (Q— F)/p, P being the sum of the least positive quadratic residues of p, and Q that of the non-residues. Writing the quadratic residues in the form =*=&, l^fc^i(p — 1), let m be the number of negative terms —A;, and —T their sum. Since —1 is a non-residue, m is the number of non-residues < f p and
Since p = 4n-}~3, A7" = n+l— m — $.   But n+1 and $ are of the same parity since
1MSitzungs. Berlin Math. Gesell., 15, 1915, 7-8.
1ML'interm<$diaire des math., 23, 1916, 174.
u°Nouv. M6m. Ac. Berlin, 2, 1773, annSe 1771, 125; Oeuvres, 3, 432.
ulMeditat. Algebr., 1770, 218; ed. 3, 1782, 380.
11JJour. fiir Math., 6, 1830, 105.
U3/6id., 3, 1828, 407-8; Werke, 1, 107.   Cf. Lucas, Th6orie des nombres, 438; Tinterm^diaire
des math., 7, 1900, 347. 1M/6R, 9, 1832, 189-92; Werke, 6, 240-4.