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CHAP, ill]               FERMAT'S AND WILSON'S THEOREMS.                       83
n~pil .. .plh (h>2), n7*2p*.   Then a system of residues modulo n, each prime to n, is given by 2J Afc, with
where r,- ranges over a system of residues modulo pf, each prime to p{. Let P be the product of these S^r,-.   Since AtAj is divisible by n if i^j,
h                           *>(n/pi«A
psSA,'w(IIr,r     ; (modn).
Thus P — 1 is divisible by each p lk and hence by n.
*Illgner147 proved Fermat's theorem.
A. Bottari148 proved Wilson's theorem by use of a primitive root [Gauss30].
J. Schumacher149 reproduced Cayley's101 proof of Wilson's theorem.
A. AreValo150 employed the sum Sn 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 — l. In the notation of Wronski, write ap/T for
a(a+r) . . . ja+(p-l)r[ =ap+S1ap-1r+ . . . +Sp^arv'1.
For a = r=l, we have p! = l+>Si+. . .+Sp-i, whence Sp-i^ — l (mod p), giving Wilson's theorem. Also, ap/r=ap— a-rp~l. Dividing by a and taking r=l, we have
(a+l^-^ssa"-1-! (mod p).
The left member is divisible by p if a is not. Hence we have Fermat's theorem. Another proof follows from Vandermonde's formula
(mod p),
Remove the factor a and set r = 0; we obtain Fermat's theorem.
Prompt151 gave Euler's14 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 unconventional notations.
F. Schuh162 proved Euler's theorem, the generalized Wilson theorem, and discussed the symmetric functions of the roots of a congruence for a prime modulus.
147Lehrsatz iiber xn~x, Untcrrichts Blatter fur Math. u. Naturwias., Berlin, 18, 1912, 15.
148I1 Boll. Matematica Gior. Sc.-Didat., 11, 1912, 289.
"•Zeitschrift Math.-naturwias. Untorricht, 44, 1913, 263-4.
1MRevista de la Socieclad Mat. Kapafiola, 2, 1913, 123-131.
161D6mon8trations nouvclloa cles th^urdmca de Fermat ct de Wilson, Paris, Gauthier-Villare,
1913, 18 pp.   Reprinted in I'intcrmddiaire de8 math., 20, 1913, end. 1MSuppl. de Vriend der Wiskunde, 25, 1913, 33-59, 143-159, 228-259.