Chap. Ill] FeRMAT's AND WilSON's THEOREMS. 75 
C. A. Laisant and E. Beaujeux^^ used the period ai . . .a„ of the periodic 
fraction to base B for the irreducible fraction pi/q, where q is prime to B. 
li P2,. . ., Pn are the successive remainders, 
Bpi = aiq+p2, Bp2 = a2q+P3,. . ., Bpr, = anq+Pi. 
Starting with the second equation, we obtain the period a2. . .a„ai for P2/q- 
Similarly for ps/q,. . ., Pn/q- Thus the f=(p{q) irreducible fractions with 
denominator q separate into sets of n each. Hence /=A;n. Since 5'*=!, 
B^=l (modg). 
L. Ottinger^- employed differential calculus to show that, in 
P={a+d){a+2d).. . \a+ip-l)d\ =aP-i+Ci^~V-2d+C2^-V-3d2_|__ ^ 
3=1 q~ri- 
Cr being the sum of the products of 1, 2, . . . , A; taken r at a time. Hence, if 
p is a prime, C?~^ (r = 1, . . ., p — 2) is divisible by p, and 
P=aP-i+c^-2d. ..{p-l)d (mod p). 
For a = d = l, this gives 0=l + (p — 1)! (mod p). 
H. Anton^^ gave Gauss' ^^ proof of Wilson's theorem. 
J. Petersen^^ proved Wilson's theorem by dividing the circumference of 
a circle into p equal parts, where p is a prime, and marking the points 
1, . . ., p. Designate by 12. . .p the polygon obtained by joining 1 with 2, 
2 with 3,. . ., p with 1. Rearranging these numbers we obtain new poly- 
gons, not all convex. While there are p! rearrangements, each polygon can 
be designated in 2p ways [beginning with any one of the p numbers as first 
and reading forward or backward], so that we get (p — 1)!/2 figures. Of 
these ^(p — 1) are regular. The others are congruent in sets of p, since by 
rotation any one of them assumes p positions. Hence p divides (p — 1)!/2 
-(p-l)/2 and hence (p-2)!-l. Cf. Cayley^o^. 
To prove Fermat's theorem, take p elements from q with repetitions in 
all ways, that is, in q^ ways. The q sets with elements all alike are not 
changed by a cyclic permutation of the elements, while the remaining q^ — q 
sets are permuted in sets of p. Hence p divides q^—q. [Cf. Perott,^^® 
Bricard.i"] 
F. Unferdinger^^ proved by use of series of exponentials that 
2''-(';')(^-ir+(2)(^-2r-...+(-ir(^)(2-mr 
' "Nouv. Ann. Math., (2), 7, 1868, 292-3. 
"Archiv Math. Phys., 48, 1868, 159-185. 
''Ibid., 49, 1869, 297-8. 
"Tidsskrift for Mathematik, (3), 2, 1872, 64-65 (Danish). 
"^Sitzungsberichte Ak. Wisa. Wien, 67, 1873, II, 363. 
