94 
History of the Theory of Numbers. 
[Chap. Ill 
(mod N) ioT q = N—l, but for no smaller q, is practical only if it be known 
that a small number a is a primitive root of N. 
G. Arnoux--^" gave numerical instances of the converse of Fermat's 
theorem. 
M. Cipolla^^^ stated that the theorem of Lucas^^' impUes that, if p is 
a prime and A: = 2, 4, 6, or 10, then kp-\-l is a prime if and only if 2^'^=1 
(mod kp-\-l). He treated at length the problem to find a for which a^~^ = 1 
(mod P), given a composite P; and the problem to find P, given a. In 
particular, we may take P to be any odd factor of (a^" — l)/(a^ — 1) if p 
is an odd prime not dividing a^ — 1. Again, 2^~^= 1 (mod P) for P = F^F^ ■ . . 
F„ m>n> . . .>s, if and only if 2'>m, where P, = 2^''4-l is a prime. If 
p and q = 2p — \ are primes and a is any quadratic residue of q, then a^«~^ = 1 
(mod pq) ; we may take a = 3 if p = 4n+3 ; a = 2\i p = 4n+ 1 ; both a = 2 and 
a = 3if p = 12A:+l;etc. 
E. B. Escott^^° noted that e"~^ = l (mod n) if e'' — 1 contains two or more 
primes whose product n is =1 (mod a), and gave a list of 54 such n's. 
A. Cunningham^^^ noted the solutions n = FsF^F^QF7, n = F^. . .P15, etc. 
[cf. CipoUa], and stated that there exist solutions in which n has more than 
12 prime factors. One with 12 factors is here given by Escott. 
T. Banachiewicz^^^ verified that 2^—2 is divisible by N for N composite 
and < 2000 only when N is 
341 = 11-31, 561=3-1M7, 1387 = 19-73, 1729 = 7-13-19, 1905 = 3-5-127. 
Since 2^—2 is evidently divisible by N for every N = Fk = 2^ +1, perhaps 
Fermat was thus led to his false conjecture that every Fk is a prime. 
R. D. CarmichaeP^^ proved that there are composite values of n (a 
product of three or more distinct odd primes) for which e"~^=l (mod n) 
holds for every e prime to n. 
J. C. Morehead^^ and A. E. Western proved the converse of Fermat's 
theorem. 
D. Mahnke^ (pp. 51-2) discussed Leibniz' converse of Fermat's theorem 
in the form that n is a prime if a:"~^=l (mod n) for all integers x prime to n 
and noted that this is false when n is the square or higher power of a prime 
or the product of two distinct primes, but is true for certain products of 
three or more primes, as 3-11-17, 5-13-17, 5-17-29, 5-29-73, 7-13-19. 
R. D. CarmichaeP^^ used the result of Lucas^^" to prove that a^~^ = l 
(mod P) holds for every a prime to P if and only if P — 1 is divisible by 
X(P). The latter condition requires that, if P is composite, it be a product 
of three or more distinct odd primes. There are found 14 products P of 
«8« Assoc, frang., 32, 1903, II, 113-4. 
"•Annali di Mat., (3), 9, 1903-4, 138-160. 
""Messenger Math., 36, 1907, 175-6; French transl., Sphinx-Oedipe, 1907-8, 146-8. 
"'Math. Quest. Educat. Times, (2), 1^, 1908, 22-23; 6, 1904, 26-7,55-6. 
»«Spraw. Tow. Nauk, Warsaw, 2, 1909, 7-10. 
"'Bull. Amer. Math. Soc, 16, 1909-10, 237-8. 
^Ibid., p. 2. 
"»Amer. Math. Monthly, 19, 1912, 22-7. 
