86 History of the Theory of Numbers. [Chap, hi 
where d ranges over those divisors of a^ — 1 which do not divide d° — l for 
0<t;<iV; while, in the former, Pi,. . ., Pa are the distinct prime factors of 
N, and n is prime to N. 
L. Gegenbauer^^- wrote F(o, n) in the form S/i(c^)a"'''', where d ranges 
over the divisors of n, and nid) is the function discussed in Chapter XIX 
on Inversion. As there shown, 2ju(d) =0 if n> 1. This case fix) =ijl{x) is 
used to prove the generaUzation : If the function /(x) has the property that 
2/((i) is divisible by n, then for every integer a the function l!>f{d)a'''^ is 
divisible by n, where in each sum d ranges over the divisors of n. Another 
special case, f{x) =4>{x), was noted by MacMahon.^^^ 
J. Westlund^^^ considered any ideal Am. o. given algebraic number field, 
the distinct prime factors Pi, . . . , P^ of ^, the norm n(^) of ^, and proved 
that if a is any algebraic integer, 
^nU) _^gn{A)ln{Pi) _|_2^n(^)/n(PiPj) _ 4- ( — n'^'>(^)/«(Pi. . -Pi) 
is always divisible by A. 
J. Vdlyi^^^ noted that the number of triangles similar to their nth pedal 
but not to the dih. pedal {d<n) is 
Xin) =^P{n) -^^(^) +2'A(-^) - ■ ■ ., 
Vp/ ^PlP2^ 
if Pi> P2, • • • are the distinct prime factors of n, and yp{k)=2^{2^ — \). He 
proved that x(^) is divisible by n, since if the nth pedal to ABC is the first 
one similar to ABC, a like property is true of the first pedal, . . ., (n — l)th 
pedal, so that the x(^) triangles fall into sets of n each of period n. [Note 
that x(n)=P(4, n)-F(2, n).] 
A. Axer^^^ proved the following generalization of Gegenbauer's"^ theorem: 
If G(ri, . . . , r/i) is any polynomial with integral coefficients, and if, when d 
ranges over all the divisors of n, 
2/(rf)G(ri"^. . ., rC'^)=Q (mod n) 
for a particular function G = Gq and a particular set of values Txq, . . . , r/,o, 
not a set of solutions of Gq, and for which Go is prime to n, then it holds for 
every G and every set ri, . . . , r^. 
Further Generalizations of Fermat's Theorem. 
For the generalization to Galois imaginaries, see Ch. VIII. 
For the generalization by Lucas, see Ch. XVII, Lucas,^^ Carmichael.^' 
On :x^= 1 (mod n) for x prime to n, see Cauchy,^^ Moreau,^^ Epstein,"' 
of Ch. VII. 
0. H. MitchelP'^ considered the 2* products s of distinct primes dividing 
k = pi...pf and denoted by r/A;) the number of positive integers 
X,<k which are divisible by s but by no prime factor of k not dividing s. 
i^Monatshefte Math. Phys., 11, 1900, 287-8. 
i"Proc. Indiana Ac. Sc, 1902, 78-79. 
"«Monatshefte Math. Phys., 14, 1903, 243-2.53. 
»"MonatBhefte Math. Phys., 22, 1911, 187-194. 
"»Amer. Jour. Math., 3, 1880, 300; Johns Hopkins Univ. Circular, 1, 1880-1, 67, 97. 
