82 History of the Theory of Numbers. [chap. hi 
where \k\ =p{p — l) . . .(p — k). Hence, if p is a prime and n<p — 1, 
Sn+i—Sn—0. But Si^O. Hence s„=0(n<p — 1), Sp_i= — (p — 1)!. Thus 
Wilson's theorem follows from Fermat's. 
Without giving references, Aubry (p. 298) attributed Horner's" proof 
of Euler's theorem to Gauss; the proof (pp. 439-440) by Paoli^^ (and 
Thue^^^) of Fermat's theorem to Euler^^; the proof (p. 458) by Laplace^^ of 
Euler's theorem by powering to Euler. 
R. D. CarmichaeP^^ noted that, if L is the 1. c. m. of all the roots z of 
0(2) = a, and if a: is prime to L, then a:°= 1 (mod L). Hence except when n 
and n/2 are the only numbers whose </)-function is the same as that of n, 
^•pM = ^ holds for a modulus M which is some multiple of n. A practical 
method of finding M is given. 
R. D. CarmichaeP^^ proved the first result by Lucas.^^° 
J. A. Donaldson^^° deduced Fermat's theorem from the theory of 
periodic fractions. 
W. A. Lindsay"^ proved Fermat's theorem by use of the binomial 
theorem. 
J. I. Tschistjakov"^ extended Euler's theorem as had Lucas. ^^° 
P. Bachmann^^^ proved the remarks by Lucas,^^° but replaced <f>+(r<n 
by 71^0 +(7, stating that the sign is > if n is divisible by at least two distinct 
primes. 
A. Thue^^ noted that a different kinds of objects can be placed into n 
given places in o" ways. Of these let 11'^ be the number of placings such 
that each is converted into itself by not fewer than n applications of the 
operation which replaces each by the next and the last by the first. Then 
U2 is divisible by n. If n is a prime, 1/1 = a""— a and we have Fermat's 
theorem. Next, a''=2[/a, where d ranges over the divisors of n. Finally, 
if p, 5, . . . , r are the distinct prime factors of n, 
C/^=S(-l)?a"/^=0 (mod n), 
where D ranges over the distinct divisors oi pq. . .r, while is the number 
of prime factors of D. Euler's theorem is deduced from this. 
H. C. PockUngton^^^ repeated Bricard's^^^ proof. 
U. Scarpis^^^ proved the generalized Wilson theorem by a method similar 
to Arndt's.^° The case of modulus 2^" (X>2) is treated by induction. 
Assume that Ilr=l (mod 2^), where ri, . . ., r„ are the v=4>{2^) odd integers 
<2^. Then rj, . . ., r„ ri+2^, . . ., r„+2'' are the residues modulo 2^"+^ and 
their product is seen to be =1 (mod 2''"''^). Next, let the modulus be 
"«BuU. Amer. Math. Soc, 15, 1908-9, 221-2. 
"'/bid., 16, 1909-10, 232-3. 
""Edinburgh Math. Soc. Notes, 1909-11, 79-84. 
"i/bwf., 78-79. 
"^Tagbl. XII Vers. Russ. Nat., 124, 1910 (Russian). 
i«Niedere Zahlentheorie, II, 1910, 43-44. 
'*^Skrifter Videnskaba-Selskabet, Christiania, 1910, No. 3, 7 pp. 
>«Nature, 84, 1910, 531. 
i«Periodico di Mat., 27, 1912, 231-3. 
