Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 275 
A. Fleck"^ proved that 
if and only if p is a prime. The case a = 1 is Wilson's theorem. 
Gu^rin^'^ asked if Wolstenholme's^^ result is new and added that 
(iLij — ^~1 (modp^), p prime >3. 
The Congruence 1-2-3. . .(p— 1)/2= ±1 (mod p). 
J. L. Lagrange^^" 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. . .P=iy^(-1)¥ (mod p). 
For p a prime of the form 4n+3, he noted that 
(5) l-2-3...^==tl (modp). 
E. Waring^^^ and an anonymous writer^^^ derived (4) in the same manner. 
G. L. Dirichlet^^^ noted that, since —1 is a non-residue of p = 4n+3, the 
sign in (5) is -f 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, 
l-2.3...^=(-ir (modp). 
C. G. J. Jacobi^^^ observed that, for p>3, m is of the same parity as N, 
where 2N—l = {Q—P)/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 ^k, l^A;^|(p — 1), let m be the number of negative 
terms —k, and — T their sum. Since — 1 is a non-residue, m is the number 
of non-residues < ^p and 
^{=i=k)=Sp, P=2{+k)-{-X{p-k)=mp+Sp, 
Since p = 4n+3, N = n-\-l—m—S. But 7i-|-l and S are of the same parity 
since 
p>S+2r = l+2+...+Kp-l)=iy-l) = (2n-H)(n+l). 
lo^Sitzungs. BerUn Math. GeseU., 15, 1915, 7-8. 
lo^L'intermgdiaire des math., 23, 1916, 174. 
""Nouv. M6m. Ac. BerHn, 2, 1773, ann6e 1771, 125; Oeuvres, 3, 432. 
"iMeditat. Algebr., 1770, 218; ed. 3, 1782, 380. 
i"Jour. fur Math., 6, 1830, 105. 
"'/bid., 3, 1828, 407-8; Werke, 1, 107. Cf. Lucas, Th^orie des nombres, 438; rinterm^diaire 
des math., 7, 1900, 347. 
i"/6id., 9, 1832, 189-92; Werke, 6, 240-4. 
