Chap. Ill] FeEMAT's AND WiLSON's ThEOKEMS. 83 
n = pi' . . .p^'* {h>2), n9^2p^. Then a system of residues modulo n, each 
h 
prime to n, is given by S A,r„ with 
i=l 
A 
' w) 
n \</>(pi°0 
} 
where r^ ranges over a system of residues modulo pi"', each prime to p,. 
Let P be the product of these SA,rj. Since AiAj is divisible by n if iT^j, 
h <p(n/pi°-i\ 
p_2Ai^W(nr,)^ ^ (modn). 
t=i 
Thus P— 1 is divisible by each p°^ and hence by n. 
*Illgner^^^ proved Fermat's theorem. 
A. Bottari^^^ proved Wilson's theorem by use of a primitive root [Gauss^°]. 
J. Schumacher^^^ reproduced Cayley's^°^ proof of Wilson's theorem. 
A. Arevalo^^° employed the sum /S„ of the products taken n at a time of 
1, 2, . . . , p — 1. By the known formula 
it follows by induction that Sn is divisible by the prime p if n<p — 1. In 
the notation of Wronski, write a^^*" for 
a{a+r).. . \a+ip-l)r\ =a''-\-Sia''-''r+ . . .+Sp_iar^-\ 
For a = r = l, we have p! = l+*Si+. . .-\-Sp^i, whence >Sp_i= — 1 (mod p), 
giving Wilson's theorem. Also, a^^''=a^—a'r^~^. Dividing by a and taking 
r = l, we have 
(a+l)(^-^^/^=a^-i-l (modp). 
The left member is divisible by p if o is not. Hence we have Fermat's 
theorem. Another proof follows from Vandermonde's formula 
(x+ay^'= S (J)x^p-''^^'a^^'=x''^'+a''^' (mod p), 
(xi + . . . +xy'-=x,^/'+ . . . +a^/^ a^/''=a-P^ 
Remove the factor a and set r = 0; we obtain Fermat's theorem. 
Prompt^^^ gave Euler's^^ proof of his theorem and two proofs of the type 
sketched by Gauss of his generalization of Wilson's theorem; but obscured 
the proofs by lengthy numerical computations and the use of unconven- 
tional notations. 
F. Schuh^^^ proved Euler's theorem, the generalized Wilson theorem, 
and discussed the symmetric functions of the roots of a congruence for a 
prime modulus. 
"^Lehrsatz uber x"— x, Uaterrichts Blatter fiir Math. u. Naturwisa., Berlin, 18, 1912, 15. 
'"II Boll. Matematica Gior. Sc.-Didat., 11, 1912, 289. 
'"Zeitschrift Math.-naturwiss. Unterricht, 44, 1913, 263-4. 
""Revista de la Sociedad Mat. Espafiola, 2, 1913, 123-131. 
"'Demonstrations nouvelles des th^orlmes de Fermat et de Wilson, Paris, Gauthier-Villars, 
1913, 18 pp. Reprinted in Tinterm^diaire des math., 20, 1913, end. 
"^Suppl. de Vriend der Wiskunde, 25, 1913, 33-59, 143-159, 228-259. 
