Chap. Ill] GENERALIZATIONS OF FeRMAT's THEOREM. 85 
Ed. Wey^^^^ E. Lucas^^^ and Pellet^^^ gave direct proofs that F{a, N) is 
divisible by N for any integers a, N. 
H. Picquet^^^ noted the divisibility of F{Zm — l, N) by iV in an enumera- 
tion of certain curvilinear polygons of N sides, at the same time inscribed 
and circumscribed in a given cubic curve. He. gave a proof of the divisi- 
bility of F{a, N) by N, requiring various subcases. He stated that the 
function F{a, N) is characterized by the two relations 
(2) F{a, np") ^F{a''\ n) -F{a''"\ n), F^a, p')=a^'-a''"\ 
where a is any integer, n an integer not divisible by the prime p. 
A. Grandi^®^ proved that F{a, N) is divisible by N by writing it as 
^N _ ^N/p, _ j ^^N/p, _ ^Nlp,p,-^ j^ ^^N/p, _ ^N/p,v,^ + . . . [ 
+ \ {a^/p^P'-a^/PiP^p>) -f- ... J -I- ... . 
Each of these binomials is divisible by pi' since 
G. Koenigs^^* considered a uniform substitution z' =4>{z) and its nth 
power z" =4)n{z). Those roots of 2— 0^(2) =0 which satisfy no like equation 
of lower index are said to belong to the index n. If x belongs to the index 
n, so do also (^i{x) for i—\,..., n — 1. Thus the roots belonging to the 
index n are distributed into sets of n. If a is the degree of the polynomials 
in the numerator and denominator of 0(s), the number of roots belonging 
to the index n is F{a, n), which is therefore divisible by n. 
MacMahon's^^^ paper contains in a disguised form the fact that F(a, N) 
is divisible by N. Proofs were given by E. Maillet"^ by substitution 
groups, and by G. Cordone.^^^ 
Borel and Drach^'^" made use of Gauss' result that F{p, N) is divisible 
by N for every prime p and integer N, and Dirichlet's theorem that there 
exist an infinitude of primes p congruent modulo N to any given integer a 
prime to N, to conclude that F(a, N) is divisible by N. 
L. E. Dickson^^^ proved by induction (from k to k-{-l primes) that 
F{a, N) is characterized by properties (2) and concluded by induction that 
F{a, N) is divisible by N. A like conclusion was drawn from 
\F{a, N)\'-F{a, N)=Fia, qN) (mod q), 
where g is a prime. He gave the relations 
F{a, nN) = Fia"", n) - i F(a^/^-, n) + S Fia"^^"'"', n)-... 
+ {-TyF{a^^'''-'", n)l 
F{a,N)=i:<l>id), 
"Casopis, Prag, 11, 1882, 39. 
""Comptes Rendus Paris, 96, 1883, 1300-2. 
»«/6td., p. 1136, 1424. Jour, de I'^cole polyt., cah. 54, 1884, 61, 85-91. 
i"Atti R. Istituto Veneto di Sc, (6), 1, 1882-3, 809. 
i"Bull. des sciences math., (2), 8, 1884, 286. 
"•Rivista di Mat., Torino, 5, 1895, 25. 
""Introd. th^orie dea nombres, 1895, 50. 
"^Annals of Math., (2), 1, 1899, 35. Abstr. in Comptes Rendus Paris, 128, 1899, 1083-6. 
