Chap. XVII] Recuering Series; Lucas' t/„, v^. 405 
Un+5=Un (mod 2), w„+5=3w„ (mod 5), ^n+6o=Wn (mod 10), 
so that Uq, Us, Uq, Ug,. . . alone are even, Uq, %, u^q, . . . are multiples of 5. 
J. Wasteels'^^ proved that two positive integers x, y, for which y^—xy—x^ 
equals +1 or — 1, are consecutive terms of the series of Pisano. If 5x^±4 
is a square, a; is a term of the series of Pisano. These are converses of 
theorems by Lucas. ^^ 
G. Candido^^ treated Un, v^, by algebra and function- theory. 
E. B. Escott^^ proved the last result in Lucas' paper. ^° 
A. Arista'^^ expressed S"if it~^ in finite form. 
M. Cipolla'^^ gave extensive references and a collection of known formulas 
and theorems on w„, v„. His apphcation to binomial congruences is given 
under that topic. 
G. Candido^^ gave the necessary and sufficient conditions, involving 
the i/j, that a polynomial x has the factor x^ — Px-\-Q, whose roots are a, b. 
A. Laparewicz'^^ treated the factoring of 2'"='= 1 by Lucas' method.^^ 
E. B. Escott^^" 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. Escott^^ applied Lucas' theory to the case it„ = 2tt„_i-f-w„_2. 
L. E. Dickson'^^" proved that if Zj, is the sum of the kth powers of the roots 
of a"'-]-pid^~^+ . . . +Pm = 0, where the p's are integers and pi = 0, then, in 
the series defined by Zx+m-\-piZx+m-i+ • • • +PmZx = 0, Zt is divisible hy i\it is 
a prime. 
E. Landau^" proved theorems on the divisors of V^, Y^, where 
{x^iT= VJx)^iYSx), i= \/^. 
P. Bachmann^^ treated at length recurring series. 
C. Ruggieri^^ used Pisano's series for w_„ to solve for ^ and r\ 
E. Zeuthen^^ proposed a problem on the series of Pisano. ^ 
H. Mathieu^^ noted that in 1, 3, 8, . . ., x,i+i = 3a:„— a:„_i, the expressions | 
a^n^n+i + 1, Xn-iXn+i-\-l are squares. — 
Valroff^^ stated in imperfect form theorems of Lucas. 
A. Aubry^^ gave a summary of results by Genocchi^^ and Lucas. 
"Mathesis, (3), 2, 1902, 60-62. 
"Periodico di Mat., 17, 1902, 320-5; I'interm^diaire des math., 23, 1916, 175-6. 
T*L'mterm6diaire des math., 10, 1903, 288. "Giornale di Mat., 42, 1904, 186-196. 
"Rendiconto Ac. Sc. Fis. e Mat. Napoh, (3), 10, 1904, 135-150. 
"Periodico di Mat., 20, 1905, 281-285. 
"Wiadomosci Matematyczne, Warsaw, 11, 1907, 247-256 (Polish). 
78oThe 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. 
"L'interm6diaire des math., 15, 1908, 248-9. "OAmer. Math. Monthly, 15, 1908, 209. 
soHandbuch. . .Verteilung der Primzahlen, I, 1909, 442-5. 
"Niedere Zahlentheorie, II, 1910, 55-96, 124. s^pgrjodico di Mat., 25, 1910, 266-276. 
"Njrt Tidsskr. for Math., Kjobenhavn, A 22, 1911, 1-9. Solution by Fransen and Damm. 
»*L'interm6diaire des math., 18, 1911, 222; 19, 1912, 87-90; 23, 1916, 14 (generalizations). 
^Ihid., 19, 1912, 145, 212, 285. "L'enseignement math., 15, 1913, 217-224 
