CHAP. Ill] CONVEKSE OF FERMAT's THEOKBM. 93 G. Levi114 was of the erroneous opinion that P is prime or composite according as it is or is not a divisor of 10P""1~1 [criticized by Cipolla,229 p. 142]. K. Zsigmondy218 noted that, if q is a prime =1 or 3 (mod 4), then 2q+l is a prime if and only if it divides (2a-f l)/3 or 2fl— 1, respectively; 4g-f-l is a prime if and only if it divides (22a+l)/5. E. B. Escott219 noted that Lucas'214 condition is sufficient but not necessary. J. H. Jeans220 noted that if p, q are distinct primes such that 2* =2 (mod g), 2*=2 (mod p), then 2P8=2 (mod pq), and found this to be the case for pq= 11-31, 19-73, 17-257, 31-151, 31-331. He ascribed to Kossett the result 2n~"1=l (mod ri) for w=645. A. Korselt221 noted this case 645 and stated that ap=a (mod p) if and only if p has no square factor and p — 1 is divisible by the 1. c. m. of px — 1, . . . , pn — 1, where pi9 . . . , pn are the prime factors of p. J. Franel222 noted that 2pfl=2 (mod pq), where p, q are distinct prunes, requires that p — 1 and #—1 be divisible by the least integer a for which 2a= 1 (mod pq) . [Cf . Bouniakowsky.213] L. Gegenbauer222a noted that 2pfl~1s=l (mod pq) if p = 2r-l = /cpr+l and q=Kr+l are primes, as for p = 31, g=ll. T. Hayashi223 noted that 2n-2 is divisible by n= 11-31. If odd primes p and q can be found such that 2P=2, 2a=2 (mod pq), then 2pfl-2 is divisible by pq. This is the case if p — 1 and q — 1 have a common factor pf for which 2^1 (mod pq), as for p = 23, g=89, p' = ll. • Ph. Jolivald224 asked whether 2N~1=1 (mod N) if Ar=2p-l and p is a prime, noting that this is true if p = ll, whence ^=2047, not a prime. E. Malo225 proved this as follows: =l (modN). G. Ricalde226 noted that a similar proof gives aN~a+l=l (mod N) if jV" = ap— 1, and a is not divisible by the prime p. H. S. Vandiver227 proved the conditions of J. Franel222 and noted that they are not satisfied if a< 10. Solutions for a = 10 and a = 11 are pq = 11-31 and 23-89, respectively. H. Schapira228 noted that the test for the primality of N that aa=l *8Monatshefte Math. Phys., 4, 1893, 79. *19L'interm6diaire des math., 4, 1897, 270. "'Messenger Math., 27, 1897-8, 174. ^L'interme'diaire des math., 6, 1899, 143. ™Ibid., p. 142. *»0Monatshefte Math. Phys., 10, 1899, 373.