94 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. Ill (mod N) for 5=^—!, but for no smaller qy is practical only if it be known that a small number a is a primitive root of N. G. Arnoux2280 gave numerical instances of the converse of Fermat's theorem. M. Cipolla229 stated that the theorem of Lucas217 implies that, if p is a prime and & = 2, 4, 6, or 10, then kp+l is a prune if and only if 2*p=l (mod kp+1). He treated at length the problem to find a for which a^"1 s 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 (a2p—l)/(a2 — 1) it p is an odd prime not dividing a2—1. Again, 2P~1=1 (mod P) for P = FmFn... FS) m>n> .. .>s, if and only if 2s>m, where Fv = 22V+l is a prime. If p and q=2p — 1 are prunes and a is any quadratic residue of q, then apfl~1=1 (mod pq)'} we may take a = 3 if p = 4n+3; a = 2 if p=4n+l; both a=2 and a = 3if p = 12fc+l;etc. E. B. Escott230 noted that en~l=I (mod ri) if ea—1 contains two or more primes whose product n is =1 (mod a), and gave a list of 54 such n's. A. Cunningham231 noted the solutions n=F3F^F5F^F7t n=F4.. .F15, etc. [cf. Cipolla], 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. Banachiewicz232 verified that 2^—2 is divisible by N for N composite and <2000 only when N is 341 = 11-31, 561 = 3-11-17, 1387 = 19-73, 1729 = 7-13-19, 1905 = 3-5-127.^ Since 2^-2 is evidently divisible by N for every N = Fk = 22k+l, perhaps Fermat was thus led to his false conjecture that every Fk is a prune. R. D. Carmichael233 proved that there are composite values of n (a product of three or more distinct odd primes) for which en~l=\ (mod n) holds for every e prime to n. J. C. Morehead234 and A. E. Western proved the converse of Fermat's theorem. D. Mahnke7 (pp. 51-2) discussed Leibniz' converse of Fermat's theorem in the form that n is a prime if xn~l^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. Carmichael235 used the result of Lucas110 to prove that ap~x = 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 atoAssoc. frang., 32, 1903, II, 113-4. »«Annali di Mat., (3), 9, 1903-4, 138-160. 230Messenger Math., 36, 1907, 175-6; French transl., Sphinx-Ocdipe, 1907-8, 146-8. 231Math. Quest. Educat. Times, (2), 14, 1908, 22-23; 6, 1904, 26-7, 55-6. 232Spraw. Tow. Nauk, Warsaw, 2, 1909, 7-10.