276 History of the Theory of Numbers. [Chap, ix 
He stated empirically that N is the number of reduced forms ay^+hyz-\-cz^, 
4:ac — b~ = p for b odd, ac — \b~ = p for b even, where b<a, b<c. 
C. F. Arndt^^^ proved in two ways that the product of all integers 
relatively prime to M = %r or 2/)", and not exceeding (ilf — 1)/2, is =±1 
(mod M), when p is a prime 4fe+3, the sign being + or — according as the 
number of residues >M/2 of M is even or odd. Again, 
{l-3-5-7...(p-2)}2=±l (modp), 
the sign being + or — according as the prime p is of the form 4n+3 or 
472+1. In the first case, 1-3. . .(p — 2)=±1 (mod p). 
L. Kronecker^^^ obtained, for Dirichlet's^^^ exponent m, the result m=j/ 
(mod 2), where v is the number of positive integers of the form q^^^'^r^ in 
the set p — 2^, p — 4", p — 6", . . ., and g is a prime not dividing r. Liou- 
\'ille (p. 267) gave m=k-\-v" (mod 2), when p = 8^+3 and v" is the number 
of positive integers of the form g'^'+V^ in the set p— 4^, p — 8^, p — 12^, . . .. 
J. Liou\'ille^" gave the result ?n=cr+r (mod 2), for the case p = 8^'+3, 
where r is the number of positive integers of the form 2g^'"^^ r^ {q a prime not 
dividing r) in the set p — 1^, p— 3^, p — 5^, . . . , and a is the number of equal 
or distinct primes 4gr+l di\'iding b, where p = a^+26" (uniquely). 
A. Korkine^^^ stated that, if [x] is the greatest integer ^x, 
_p-3 
(p-3)/4 
S [Vp^l (modp). 
4 
J. Franel"^ proved the last result by use of Legendre's symbol and 
(-i)""'TG> ©=(-^)'' "-TBI (-°'^2)- 
M. Lerch^^° obtained Jacobi's"* result. 
H. S. Vandiver^^"" proved Dirichlet's"^ result and that 
(p-i)/2r.-2-i 
m= S Y~\ (mod 2). 
R. D. CarmichaeP^^ noted that (4) holds if and only if p is a prime. 
E. Malo^^^ considered the residue ±r of 1-2. . .(p — 1)/2 modulo p, 
where p is a prime 4m+l, and 0<r<p/2. Thus r^= —1. The numbers 
2, 3, . . ., (p — 1)/2, with r excluded, may be paired so that the product 
of the two of a pair is = =•= 1 (mod p) . If this sign is minus for k pairs, 
1-2. . .(p-l)/2=(-l)V (mod p). 
*J. Ouspensky gave a rule to find the sign in (5). 
Other Congruences Involving Factorials. 
V. Bouniakowskyi29 noted that (p-l)! = PP', P±P'=0 (mod p) accord- 
ing as p = 4/j=f1. For, if p is a primitive root of p, we may set P = pp^ 
I'SArchiv Math. Phys., 2, 1842, 32, 34-35. i^o^Amer. Math. Monthly, 11, 1904, 51-6. 
"«Jour. de Math., (2), 5, 1860, 127. ^^'IMd., 12, 1905, 106-8. 
^^Ubid., 128. '22i;interm6diaire des math., 13, 1906, 131-2 
»«L'interm6diaire des math., 1, 1894, 95. »23Bu11. Soc. Phys. Math. Kasan, (2), 21. 
"»/6id., 2, 1895, 35-37. i"M6m. Ac. Sc. St. P6tersbourg, (6), 1, 1831, 564. 
'"Prag Sitzungsber. (Math.). 1898, No. 2. 
