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CHAP, xvii]               RECURRING SERIES; LUCAS' un, vn.                      405
5SMn (mod 2),          ttn+5=3wn (mod 5),          wn+60=wn (mod 10),
so that i/o, y>z, u6, u9, . . . alone are even, w0, ^5, ^io> • • • are multiples of 5.
J. Wasteels72 proved that two positive integers x, y, for which y2—xy—x2 equals +1 or — I, are consecutive terms of the series of Pisano. If 5x2=±=4 is a square, x is a term of the series of Pisano. These are converses of theorems by Lucas.17
G. Candido73 treated un, vn, by algebra and function-theory.
E. B. Escott74 proved the last result in Lucas' paper.40
A. Arista75 expressed S^w'1 in finite form.
M. Cipolla76 gave extensive references and a collection of known formulas and theorems on un, vn. His application to binomial congruences is given under that topic.
G. Candido77 gave the necessary and sufficient conditions, involving the ut) that a polynomial x has the factor x2— fx+Q, whose roots are a, 6.
A. Laparewicz78 treated the factoring of 2m=1= 1 by Lucas' method.39
E. B. Escott78a showed the connection between Pisano' s series and the puzzle to convert a square into a rectangle with one more (or fewer) units of area than the square.
E. B. Escott79 applied Lucas' theory to the case un=*2un_i~\-un^2.
L. E. Dickson79* proved that if zk is the sum of the kth powers of the roots of am+piam~1+ . . . +pm = 0, where the p's are integers and pi = 0, then, in the series denned by zx+m+PiZx+m-i+ • - • +PmZ*=0, z< is divisible by t if t is a prime. ,
E. Landau80 proved theorems on the divisors of Um, Vm, where (x+i)m = Vm(x) +iVm(x\ i = V~[.
P. Bachmann81 treated at length recurring series.
C. Ruggieri82 used Pisano's series for u^n to solve for £ and 77
E. Zeuthen83 proposed a problem on the series of Pisano. H. Mathieu84 noted that in 1, 3, 8, . . ., xn+i = 3xn— £„_!, the expressions Zn4-i+l, x^iXn+i+l are squares. Valroff 85 stated in imperfect form theorems of Lucas. A. Aubry86 gave a summary of results by Genocchi15 and Lucas.
"Mathesis, (3), 2, 1902, 60-62,
73Periodico di Mat., 17, 1902, 320-5; I'intermSdiaire des math., 23, 1916, 175-6. 74L'iiiterm<§diaire des math., 10, 1903, 288.          "Giornale di Mat., 42, 1904, 186-196.
7flRendiconto Ac. Sc. Fie. e Mat. Napoli, (3), 10, 1904, 135-150. "Periodico di Mat., 20, 1905, 281-285.
78Wiadomosci Matematyczne, Warsaw, 11, 1907, 247-256 (Polish).
78aThe Open Court, August, 1907.    Reproduced by W. F. White, A Scrap-Book of Elementary Mathematics, Notes, Recreations, Essays, The Open Court Co., Chicago, 1908, 109-113. 79L'interme'diaire des math., 15, 1908, 248-9.          79aAmer. Math. Monthly, 15, 1908, 209.
fl°Handbuch. . .Verteilung der Primzahlen, I, 1909, 442-5.
81Niedere Zahlentheorie, II, 1910, 55-96, 124.           "Periodico di Mat., 25, 1910, 266-276.
MNyt Tidsskr. for Math., Kjobenhavn, A 22, 1911, 1-9.   Solution by Fransen and Damm. ML'interm6diaire des math., 18, 1911, 222; 19, 1912, 87-90; 23, 1916, 14 (generalizations). KIbid., 19, 1912, 145, 212, 285.                                88L'enseignement math., 15, 1913, 217-224