384 History of the Theory of Numbers. [Chap. XVI 
Lucas" gave the factors of 2'"+! for w =4n^60 and for 72, 84; also for 
rM=4n+2^102 and for 110, 114, 126, 130, 138, 150, 210. 
E.Catalan=« noted that x^+2{q-r)x^-^q~ for x^ ^ (2r)2*+i has the rational 
factors (2r)^*'*"^ =*= (2r)*"'"^ +5. The case r = q = \ gives LeLasseur's^^ formula. 
Again, 3'*+' + l has the factors ^^'^' + \, 32'^+^±3*+^ + l. 
S. R^alis-^'' deduced LeLasseur's'-^ formula and 24"+22"+l=n(22''± 
2"+l). 
J. J. Sylvester'^^ considered the cyclotomic function ypt{x) obtained by 
setting a + a"' =x in the quotient by a*""^ of 
(a'-l)n(a'^''''''-l) 
(1) ^.(a)= ^" n(a'^'-l). (^ = P^'- • P"'"^' 
where Pi,- ■ ■, Pn are distinct primes. He stated that every di\'isor of i/'t(x) 
is of the form kt^ 1, with the exception that, if t = p\p=^l)/m, p is a divisor 
(but not p^). Conversely, every product of powers of primes of the form 
kt^l is a divisor of i/',(x). Proofs were given by T. Pepin, ibid., 526; E. 
Lucas, p. 855; Dedekind, p. 1205 (by use of ideals). Lucas added that 
p = 2^''+3 — 1 and p = 2^^''"''^ — 1 are primes if and only if they divide </'p+i(x) 
for x = \/^ and x = 3\/^, respectively. 
A. Lefebure^° determined poljTiomipls having no prime factor other than 
those of the form HT+1, where H is given. First, let T = n\ where n is a 
prime. For A, B relatively prime integers, 
/ln_ Dn 
has, besides n, no prime factor except those of the form Hn*+1, when A 
and B are exact n'~Hh powers of integers. Second, let T = n'm^, where n, m 
are distinct primes. The integral quotient of F^iu'", v'") by F„(w, t') has only 
prime factors of the form Hn^m^ + l if u, v are powers of relatively prime 
integers with the exponent w''~^n'~^ Similarly, if T is a product of powers 
of several primes. 
Lef^bure^^ discussed the decomposition into primes of U^ — V^, where 
U, V are powers whose exponents involve factors of R. 
E. Lucas^- stated that if n and 2/1 + 1 are primes, then 2n + l is a factor 
of 2** — 1 or 2"+l according as n=3 or n=l (mod 4). If n and 4n + l are 
primes, 4n + l is a factor of 2^''+l. If n and Sn + l =.4^ + 165^ are primes, 
then 8n + l is a factor of 22" + 1 if B is odd, of 2-"± 1 if 5 is even. Also ten 
theorems stating when Qn + l=U.^+SM^, 12n + l=L^-\-12M^ or 24n + l 
= L'^-\-4SM^ are prime factors of 2'"'±1 for certain k's. 
*'Sur la s^rie r^currente de Fermat, Rome, 1879, 9-10. Report by Cunningham.*' 
"Aesoc. fran^. avanc. sc, 9, 1880, 228. 
»»^Nouv. Ann. Math., (2), 18, 1879, 500-9. 
"Comptes Rendus Paris, 90, 1880, 287, 345; Coll. Math. Papers, 3, 428. Incomplete in Math. 
Quest. Educ. Times, 40, 1884, 21. 
"Ann. 8C. 6cole norm, sup., (3), 1, 1884, 389-404; Comptea Rendus Paris, 98, 1884, 293, 413, 
567, 613. 
".\nn. BC. 6cole norm, sup., (3), 2, 1885, 113. 
"Assoc, franc, avanc. sc, 15, 1886, II, 101-2. 
