Chap. Ill] CONVERSE OF FeRMAt's THEOREM. 93 
G. Levi^^* was of the erroneous opinion that P is prime or composite 
according as it is or is not a divisor of 10^"^ — 1 [criticized by Cipolla,^^^ 
p. 142]. 
K. Zsigmondy^^^ noted that, if g is a prime =1 or 3 (mod 4), then 2q+l 
is a prime if and only if it divides (2*+l)/3 or 2^ — 1, respectively; 4g+l is 
a prime if and only if it divides (2^*+l)/5. 
E. B. Escott^^^ noted that Lucas'^^^ condition is sufficient but not 
necessary. 
J, H. Jeans^^® noted that if p, q are distinct primes such that 2^=2 
(mod g), 2^=2 (mod p), then 2^^ =2 (mod pq), and found this to be the case 
for pg = 11-31, 19-73, 17-257, 31-151, 31-331. He ascribed to Kossett the 
result 2"-^=l (mod n) for n = 645. 
A. Korselt^^^ noted this case 645 and stated that a^=a (mod p) if and 
only if p has no square factor and p — 1 is divisible by the 1. c. m. of pi — 1, . . . , 
p„ — 1, where pi, . . ., p„ are the prime factors of p. 
J. FraneP^^ noted that 2^^ =2 (mod pq), where p, q are distinct primes, 
requires that p — 1 and g — 1 be divisible by the least integer a for which 
2''=1 (mod pq). [Cf. Bouniakowsky.^^^] 
L. Gegenbauer222« noted that 2^''-^=l (mod pq) if p = 2'-l = /cpr+l 
and q = KT-\-l are primes, as for p = Sl, q = ll. 
T. Hayashi^^^ noted that 2''— 2 is divisible by n = 11-31. If odd primes 
p and q can be found such that 2^=2, 2^=2 (mod pq), then 2^'— 2 is divisible 
by pq. This is the case if p — 1 and q — 1 have a common factor p' for which 
2"''=! (mod pq), as for p = 23, g = 89, p' = ll. 
Ph. Jolivald224 asked whether 2^-^=1 (mod N) if N = 2''-l and p is a 
prime, noting that this is true if p = ll, whence iV = 2047, not a prime. 
E. Malo^^^ proved this as follows: 
AT-l =2(2^-1-1) =2pw, 2^-^ = (2^)2- = (Ar+l)2-=i (modiV). 
G. Ricalde^^^ noted that a similar proof gives a^~''+^=l (mod N) if 
N = a^—1, and a is not divisible by the prime p. 
H. S. Vandiver^^^ proved the conditions of J. FraneP^^ and noted that 
they are not satisfied if a< 10. Solutions for a = 10 and a = 11 are ^3 = 11-31 
and 23-89, respectively. 
H. Schapira^^^ noted that the test for the primality of N that a^=l 
»8Monat8hefte Math. Phys., 4, 1893, 79. 
*i«L'mtenn4diaire des math., 4, 1897, 270. 
220Messenger Math., 27, 1897-8, 174. 
''"L'interm^diaire des math., 6, 1899, 143. 
^Ibid., p. 142. 
=«""Monatshefte Math. Phys., 10, 1899, 373. 
'^'Jour. of the Physics School in Tokio, 9, 1900, 143-4. Reprinted in Abhand. Geschichte Math. 
Wiss., 28, 1910, 25-26. 
*"L'interm4diaire des math., 9, 1902, 258. 
^lUd., 10, 1903, 88. 
*^Ibid., p. 186. 
*"Amer. Math. Monthly, 9, 1902, 34-36. 
»»Tchebychef's Theorie der Congruenzen, ed. 2, 1902, 306. 
