Chap. XVII] Recuering Series; Lucas' u^, y„. 403 
P1+P3+ ■ . • +P2n-X=Pn\ P2 + -P4+ • • • +P2n = PnPn+l. 
If P^=P^ (mod Px) then n=m (mod 2X). Also, 
P„=x"-+ S (_i). (n-fe-l)...(n-2fc) ^„.,,_,_ 
fc=i 1-2. . .A; 
If a„/6„ is the nth convergent to i_j+^_..., then a„+2 = ^^n+i~cf„, 6„ 
= o„+i. Hence a^^Pn if cti = 1 , 02 = a;. 
Sylvester stated and W. S. Foster^^" proved that if f{d) is a polynomial 
with integral coefficients and Ux+i=f{Uj^, Ui=f{0), and 5 is the g. c. d. of 
r, s, then Us is the g. c, d. of Ur, Ug. 
A. Schonfiies^® considered the numbers no = l, Ui,..., n^ defined by 
n^ = n^_n^-i+ri^-2_ . . . +(-1)^ (X = 0, 1,. . .) 
and proved geometrically that if n^-i is the least of these numbers which 
has a common factor with n^, then r is a divisor of g+1, while a relation 
'mni=mnr+i (mod n^) 
holds for every index i. 
L. Gegenbauer^^ gave a purely arithmetical proof of this theorem. 
E. Lucas^^ gave an exposition of his theory, with an introduction to 
recurring series. 
M. Frolov^^ used a table of quadratic residues of composite numbers to 
factor Lucas' numbers v„. 
D. F, Seliwanov^° proved Lucas' results on the factors of w„, ?;„. 
E. Catalan^^ gave the first 43 terms of the series of Pisano, noted that 
Un divides U2y,+i, that Uzn is a sum of two squares, and treated the series 
Un = aun-i + w„-2, Ux = a, W2 = a^ + L 
Fontes^^" proved theorems stated by Lucas^^ (p. 127), and found in an 
elementary way the general term of Pisano's series, as given by Binet^^. 
E. Maillet^^'' proved that a necessary condition that every positive 
integer, exceeding a certain limit, shall equal (up to a limited number of 
units) the sum of the absolute values of a finite number of terms of a recur- 
ring series, satisfying an irreducible law of recurrence with integral coeffi- 
cients, is that all the roots of the corresponding generating equation be roots 
of unity. 
W. ManteP^ noted that, if the denominator F{x) of the generating 
fraction of a recurring series is irreducible modulo p, a prime, the residues 
modulo p of the terms of the recurring series repeat periodically, and the 
length of a period is at most p" — 1 ; the proof is by use of Galois' general- 
ization of Fermat's theorem. The case of a reducible F{x) is also treated. 
65<iMath. Quest. Educ. Times, 50, 1889, 54-5. ^«Math. Annalen, 35, 1890, 537. 
"Denkschriften Ak. Wiss. Wien (Math.), 57, 1890, 528. 
68Theorie des nombres, 1891, 299-336; 30; 127, ex. 1. A pamphlet, pubhshed privately by 
Lucas in 1891, is cited in I'intermediaire des math., 5, 1898, 58. 
"Assoc, frang. avanc. sc, 21, 1892, 149. 
soMath. Soc. Moscow, 16, 1892, 469-482 (in Russian). 
"Mem. Acad. R. Belgique, 45, 1883; 52, 1893-4, 11-14. 
""Assoc, frang. avanc. sc, 1894, II, 217-221. "''Assoc, frang. avanc. sc, 1896, II, 78-89 
•^Nieuw Archief voor Wiskunde, Amsterdam, 1, 1895, 172-184. 
