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402                   HISTORY OF THE THEORY OF NUMBERS.         [CHAP, xvn
Magnon,47 in reply to Lucas, proved that
if an— 1 is the sum of the squares of the first n— 1 terms of Pisano's series.
H. Brocard48 studied the arithmetical properties of the U's defined by Un+i = Un+2Un~i> Uo=l, Ui**3, in connection with the nth pedal triangle.
E. Ces&ro49 noted that if Un is the nth term of Pisano's series, then (2l7+l)n-C73n=0, symbolically.
E. Lucas60 gave his23 test for the primality of 24a+3 — 1.
A, Genocchi51 reproduced his16 results.
M. d'Ocagne52 proved for Pisano's series that [Lucas16]
The main problem treated is that to insert p terms ai, . . . , ap between two given numbers a0= a, ap+i=6, such that at-=at-_i+at-_2«    The solution is
Most of the paper is devoted to the question of the maximum number of negative terms in the 'series of a's.
E. Catalan520 proved that Un*-Un_pUn+p= (-l)w~m#Vi for Pisano's series. "E. Lucas63 stated, apropos of sums of squares, that
t^-K+2,            v4n= (2un)2+u\+2.
L. Kronecker64 obtained Dirichlet^s9 theorems by use of modular systems. Lucas540 proved that, if un = (an-bn)/(a-b),
is divisible by up when p is a prime and n is odd and not divisible by p, and by u^ when n = 2p+l.
L. Liebetruth55 considered the series Pl = 1 , P2 = x, . . . , Pn = zPn_! — Pn_2, and proved any two consecutive terms are relatively prime, and
Taking n=2X, 3X, . . ., we see that Px is a common factor of P2X, Pax, The g. c. d. of Pm, Pn is Pd} where d is the g. c. d. of m, n.   Next,
<7Nouv. Corresp. Math., 6, 1880, 418-420.'      48Nouv. Corresp. Math., 6, 1880, 145-151. "/Wa., 528; Nouv. Ann. Math., (3), 2, 1883, 192;   (3), 3, 1884, 533.   Jornal de Sc. Math.
Astr., 6, 1885, 17.
MR6cr€ations math&natiques, 2, 1883, 230.      51Comptes Rendus Paris, 98, 1884, 411-3. "Bull Soc. Math. France, 14, 1885-6, 20-41.
62«M<§ni. soc. roy. sc. Ltege, (2), 13, 1886, 319-21 ( = M61anges Math., II). "Mathesis, 7, 1887, 207; proofs, 9, 1889, 234-5.
"Berlin Berichte, 1888, 417-423; Werke, 3, 1, 281-292.   Cf. Kronecker88 of Ch. XVI. "oAssoc. franc. . avanc. sc., 1888, II, 30.           "Beitrag zur Zahlentheorie, Progr., Zerbst, 1888.