Chap. XVII] RECURRING SeRIES; LucAS' U„, y„. 401 
Upr/Ur in the form Ax^^pQ''y'^. The prime factors of 3^^±1 are given on 
p. 280. The proper divisors of 2^"+l are known to be of the form Snq-\-l; 
it is shown that q is even. Thus for 2^^+l the first divisor to be tried is 
641, for 2^^^+! the first one is 114689; in each case the division is exact 
(cf. Ch. XV). The following is a generalization: If the product of two I 
relatively prime integers a and h is of the form 4/i+l, the proper divisors 
of a^'^^^+b^"^'^ are of the form Sahnq+1. A primality test for 2'^^+^-! isj 
given. Finally, p = 2^"^^+^"+^ — 1 is a prime if and only if 
(2"+V2^^)^+(2"-V2^H4)^ = (mod p). 
T. Pepin* ^ gave a test for the primality of g = 2" — 1. Let 
^1- ^2-1-52 (modg') 
and form the series Ui, U2y. . ., it„_i by use of 
u^+i=u^ — 2 (mod q). 
Then g is a prime if and only if u^-i is divisible by q. This test differs from 
that by Lucas^^ in the choice of Ui. 
E. Lucas"*^ reproduced his^^ test for the primality of 2^ A — 1, etc., and the 
test at the end of another paper,*" with similar tests for 2*'^+^ — 1 and 2^^*'+^ — 1 . 
G. de Longchamps*^ noted that, if dk = Uk — aUk-i, 
d, = h''-\ d^d, = ¥+^-\ 
with the generalization 
X 
Jldj,.=d„ s = pi+. . .+p^-x+l. 
Take pi = . . . = p^; = p. Hence 
[Up CLUp^i) =Upx—x+l f^Upx—x' 
There is a corresponding theorem for the v's. 
J. J. Sylvester** considered the g. c. d. of u^, u^+i if 
w^ = {2x - l)Ux-i - {x-l)u^_2' 
E. Gelin*^ stated and E. Cesaro*^ proved by use of ?7„+p= UpUn+ C/p_iC/„_i 
that, in the series of Pisano, the product of the means of four consecutive 
terms differs from the product of the extremes by ± 1 ; the fourth power of 
the middle term of five consecutive terms differs from the product of the 
other four terms by unity. 
"Comptes Rendus Paris, 86, 1878, 307-310. 
«Bull. Bibl. Storia Sc. Mat. e Fis., 11, 1878, 783-798. The further results are cited in Ch. XVI. 
Comptes Rendus, 90, 1880, 855-6, reprinted in Sphinx-Oedipe, 5, 1910, 60-1. 
«Nouv. Corresp. Math., 4, 1878, 85; errata, p. 128. 
«Comptes Rendus Paris, 88, 1879, 1297; Coll. Papers, 3, 252. 
«Nouv. Corresp. Math., 6, 1880, 384. 
"Ibid., 423-4. 
