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384                   HISTORY OF THE THEORY or NUMBERS.           [CHAP, xvi
Lucas27 gave the factors of 2m+l for w=4n^60 and for 72, 84; also for w =4^+2^102 and for 110, 114, 126, 130, 138, 150, 210.
E. Catalan28 noted that x*+2(q -r^+g2 for z2 = (2r)2*+1 has the rational factors (2r)2W=fc (2r)*+1+g. The caser=# = l gives LeLasseur's23 formula. Again, 36*+3+l has the factors 32*+1+l, 32*+1=t3*+1+l.
S. R<alis280 deduced LeLasseur's23 formula and 24n+22n+l=H(22n=fc
J. J. Sylvester29 considered the cyclotomic function \f/t(x) obtained by setting a-f a""1*^ in the quotient by a*a)/2 of
where plt . . . , pn are distinct primes. He stated that every divisor is of the form Atf=*= 1, with the exception that, if < p*(p=Fl)/m, p is a divisor (but not p2). Conversely, every product of powers of primes of the form Art=*=l is a divisor of &($). Proof s were given by T. Pepin, flM., 526; E. Lucas, p. 855; Dedekind, p. 1205 (by use of ideals). Lucas added that 39=24A+3-J_and p=212*ffB-=l are primes if and only if they divide Vvt-i(z) for z=\/ 1 and a? = 3\/ 1> respectively.
A. Lefe*bure30 determined polynomisls having no prime factor other than those of the form JBT+1, where H is given. First, let Tn*, where n is a prime. For A, J5 relatively prune integers,
has, besides n, no prime factor except those of the form ffn'+l, when A and B are exact 7i*~Hh powers of integers. Second, let !T=n'?w*, where n, m are distinct primes. The integral quotient of Fn(um, vm) by Fn(u, v) has only prime factors of the form Hntfmhjr\ if u, v are powers of relatively prime integers with the exponent ra*" V"1. Similarly, if T is a product of powers of several primes.
Lefelmre31 discussed the decomposition into primes of UR  VR, where 17, V are powers whose exponents involve factors of R.
E. Lucas32 stated that if n and 2n-fl are primes, then 2n+l is a factor of 2n  1 or 2n+l according as n=3 or n=l (mod 4). If n and 4n+l are primes, 4n +1 is a factor of 22n+l. If n and 8n-fl = A2+162 are primes, then 8n+l is a factor of 22ft+l if B is odd, of 22n 1 if B is even. Also ten theorems stating when 6n+l=4L2-f3Jlf2, 12n+l=L2+12M2 or 24n-fl =L2-f48M2 are prime factors of 2*n==l for certain k's.
S7Sur la s6rie r^currente de Fennat, Rome, 1879, 9-10.   Report by Cunningham.68
"Assoc. frang. avanc. sc., 9, 1880, 228.
28flNouv. Ann. Math., (2), 18, 1879, 500-9.
28Comptes Rendus Paris, 90, 1880, 287, 345; Coll. Math. Papers, 3, 428.   Incomplete in Math.
Quest. Educ. Times, 40, 1884, 21. 80Ann. sc. e*cole norm, sup., (3), 1, 1884, 389-404; Comptes Rendus Paris, 98, 1884, 293, 413,
567, 613.
"Ann. sc. e*cole norm, sup., (3), 2, 1885, 113. *2ABSoc. franc, avanc. sc., 15, 1886, II, 101-2.