CHAP. XVH] KECUBMNG SEEEBSJ LUCAS' Wn, Vn. 403 ! =PB2, If PB=Pm (mod PJ then tt=sOT (mod 2X). Also, 2 (- L'Zi . , . K If an/bn is the nth convergent to L|+*-- • •> then = a^. Hence on = Pn if aj. = 1, a2 = a;. Sylvester stated and W. S. Foster560 proved that if /(0) is a polynomial with integral coefficients and ux+l=f(ux)1ui=f(G)) and 5 is the g. c. d. of r, s, then us is the g. c. d. of ur, u9. A. Schonflies66 considered the numbers ti0=l, nl}. . ., .n^ denned by nx=^_7^i+^--2_ . . .-K-l)x (\=0, 1,. J .) and proved geometrically that if nr«x is the least of these numbers which has a common factor with nq, then r is a divisor of q+l, while a relation mni=mnr+i (mod nq) holds for every index i. L. Gegenbauer57 gave a purely arithmetical proof of this theorem. E. Lucasf8 gave an exposition of his theory, with an introduction to recurring series. M. Frolov59 used a table of quadratic residues of composite numbers to factor Lucas* numbers vn. D. F. Seliwanov60 proved Lucas' results on the factors of u~. v~. E. Catalan61 gave the fir Un divides 172 +i>. that U2n is Fontes610 proved theorems stated bj elementary way the general term of Pk E. Mattlet615 proved that a necessai^ „ K _______ , 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 recurring series, satisfying an irreducible law of recurrence with integral coefficients, is that all the roots of the corresponding generating equation be roots of unity. W. Mantel62 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 pn — 1 ; the proof is by use of Galois' generalization of Format's theorem. The case of a reducible F(x) is also treated. 65aMath. Quest. Educ. Times, 50, 1889, 54-5. S6Math. Annalen, 35, 1890, 537. 67Denkschriften Ak. Wiss. Wien (Math.), 57, 1890, 528. 68Th6orie des nombres, 1891, 299-336; 30; 127, ex. 1. A pamphlet, published privately by Lucas in 1891, is cited in I'interme'diaire des math., 5, 1898, 58. "Assoc. frang. avanc. sc., 21, 1892, 149. e°Math. Soc. Moscow, 16, 1892, 469-482 (in Russian). 61M<§m. Acad. R. Belgique, 45, 1883; 52, 1893-4, 11-14. 81aAssoc. fran$. avanc. sc., 1894, II, 217-221. fllbAfisoc. franc,, avanc. sc., 1896, II, 78-89 MNieuw Archief voor Wiskunde, Amsterdam, 1, 1895, 172-184.