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] 2 P- 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 d 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.