402 History of the Theory of Numbers. [Chap, xvii 
Magnon,'*^ in reply to Lucas, proved that 
) Oi Oj Og ■ ■ ■ 1 — V5 
I if a„ — 1 is the sum of the squares of the first n — 1 terms of Pisano's series. 
H. Brocard^^ studied the arithmetical properties of the U's defined by 
Un+i = Un-\-2U„_i, Uo = l, Ui=3, in connection with the nth pedal triangle. 
E. Ces^ro^^ noted that if Un is the nth term of Pisano's series, then 
(2C/+l)"-C/^'* = 0, symbolically. 
E. Lucas^" gave his^^ test for the primahty of 2'*^+^ — !. 
A. Genocchi" reproduced his^^ results. 
M. d'Ocagne^^ proved for Pisano's series that [Lucas^®] 
iu, = u^2-h 2)(-iyii,= (-l)X-i-l, lim-^=(i±^', 
»=0 P=« (*p-i \ Z / 
The main problem treated is that to insert p terms ai, . . ., ap between two 
given numbers ao = a, ap+i = h, such that aj = a;_i+a,_2. The solution is 
hUi-{-{ — iyaUp+i.i 
Up+i 
Most of the paper is devoted to the question of the maximum number of 
negative terms in the series of a's. 
E. Catalan^2apj.Q^3^ ^^^^ uji^Ur,-pU,+p= (-iy-'^^U\_i for Pisano's 
series. 
E. Lucas^^ stated, apropos of sums of squares, that 
L. Kronecker^^ obtained Dirichlet's^ theorems by use of modular systems. 
Lucas^" proved that, if w„=(a''— 5")/(a — 6), 
Wp_l —U(p_i)n/Un 
is divisible by Up when p is a prime and n is odd and not divisible by p, and 
by Up when n = 2p+l. 
L. Liebetruth^^ considered the series Pi = 1 , P2 = x, . . . , P^ = ^Pn-i —Pn-2) 
and proved any two consecutive terms are relatively prime, and 
Pn = PxPn-X+l-Px-lPn-X (X<n). 
Taking n = 2X, 3X, . . . , we see that Px is a common factor of P2X, Psx, • • • • 
The g. c. d. of P,„, P„ is Pj, where d is the g. c. d. of ?n, n. Next, 
«'Nouv. Corresp. Math., 6, 1880, 418-420. '"Nouv. Corresp. Math., 6, 1880, 145-151. 
"/bid., 528; Nouv. Ann. Math., (3), 2, 1883, 192; (3), 3, 1884, 533. Jornal de Sc. Math. 
Astr., 6, 1885, 17. 
»0R6cr6ation8 mathdmatiques, 2, 1883, 230. "Coraptes Rendus Paris, 98, 1884, 411-3. 
"'Bull. Soc. Math. France, 14, 1885-6, 20-41. 
^^M6m. soc. roy. sc. Li^ge, (2), 13, 1886, 319-21 ( = M61anges Math., II). 
"Mathesis, 7, 1887, 207; proofs, 9, 1889, 234-5. 
"Berlin Berichte, 1888, 417-423; Werke, 3, I, 281-292. Cf. Kronecker'" of Ch. XVI. 
""Assoc. franQ. avanc. sc, 1888, II, 30. "Beitrag zur Zahlentheorie, Progr., Zerbst, 1888. 
