Chap. XVI] FACTORS OF a**=*=6''. 389 
L. E. Dickson and E. B. Escott^^ discussed the divisibility of p^^* — 1 
by d(p"^'^ — 1), where d is a divisor of n, and d of d. 
R. D. CarmichaeF° proved that if P^^—R^" is divisible by 8a and we 
set Q = {P''—R'')/{a{P — R) } , then Q/8 is an integer if and only if a is divisible 
by the least integer e for which P^ — R^ is divisible by each prime factor of 
a not dividing P — R, and 5 is a divisor of Q. Proof for the case R = l had 
been given by E. B. Escott'^^ 
A. Cunningham^^ tabulated the factors of y^^^^l for ?/ = 2, 3, 5, 7, 12. 
K. J. Sanjana'^^ considered the factors of 
Sanjana'^^" applied his method to prove the statement of M. Kannan that 
20^5-1= 11. 19-31-61-251421-3001-261451-64008001-3994611390415801 
•4199436993616201. 
L. E. Dickson''^ factored w"— 1 for various values of n. 
R. D. CarmichaeF^ employed the methods of Dickson^ ^ to obtain general- 
izations. Let Q„(a, /3) be the homogeneous form of Fn{a), Let n = lip/*, 
where the p's are distinct primes, and let c be a divisor of n and a multiple 
of pi°'. If a, j3 are relatively prime, the g. c. d. of 5 = a"^^'— iS'*''^' and 
Qc{a, /3) is 1 or pi and at most one Qda, /3) contains the factor pi when 
d contains pi^; if pi>2 divides 5, at most one Qcio-, jS) contains pi, and no 
one of them contains pi^. If a, /3 are relatively prime and c = mpi"', where 
m>l and m is prime to pi, then QXcl, jS) is divisible by pi if and only if 
fjx—j^x ^jj^Q^ p^) holds for x = m, but not for 0<a;<m; in all other cases 
Q= 1 (mod m). If a, jS are relatively prime, QXo-, jS), and hence also a"—^", 
has a prime factor not dividing a*— j3'(s<c), except in the cases (i) c = 2, 
^ = 1^ a = 2^-l; (ii) Q,(a, /3) =p = greatest prime factor of c, and a"^^=/3"/^ 
(modp); (iii)a(a,/3) = l. 
E. Miot^^ noted that LeLasseur's^^ formula is the case m = n = l of 
/02fc+1^2\ 2 / 02t+1^2 \ 
(f_JL) +^2^n(m+^-^±2*+in 
\ m / \ m I 
Welsch (p. 213) stated that the latter is no more general than the case A: = 0, 
which follows from the known formula for the product of two sums of two 
squares. 
A. Cunningham^^ noted the decomposition into primes : 
2"+l = 3-43-617-683-78233-35532364099. 
"L'interm^diaire des math., 1906, 87; 1908, 135; 18, 1911, 200. Cf. Dickson." 
'"Amer. Math. Monthly, 14, 1907, 8-9. 
'i76id., 13, 1906, 155-6. 
"Report British Assoc, 78, 1908, 615-6. 
"Proc. Edinburgh Math. Soc, 26, 1908, 67-86; corrections, 28, 1909-10, viii. 
"aJour. Indian Math. Club, 1, 1909, 212. 
'^Messenger Math., 38, 1908, 14-32, and Dickson"*"' of Ch. XIV. 
'^Amer. Math. Monthly, 16, 1909, 153-9. 
'«L'interm6diaire des math., 17, 1910, 102. 
"Report British Assoc, for 1910, 529; Proc. London Math. Soc, (2), 8, 1910, xiii. 
