404 History of the Theory of Numbers. [Chap, xvii 
R. W. D. Christie^ stated that, for the recurring series defined by 
a„+i = 3a„ — a„_i, 27n — l is a prime if and only if a^ — 1 is di\'isible by 
2m — 1. The error of this test was pointed out by E. B. Escott.^ 
S. R^alis"" noted that two of A'' consecutive terms of 7, 13, 25, . . ., 
3(n^H-n)+7, . . . are di\'isible by iV if iV is a prime 6/^2 + 1. 
C. E. Bickmore^** discussed factors of w„ in the final series of Catalan^\ 
He^" and others gave known formulas and properties of Pisano's series. 
R. Perrin®^ employed r„ = r„_2 + r„_3, ro = 3, ^'l = 0, ?'2 = 2. Then v^ is 
di\'isible by n if n is a prime. This was verified to be not true when n is 
composite for a wide range of values of n. The same subject was considered 
by E. jMalo^® and E. B. Escott®" who noted that Perrin's test is incomplete. 
SeveraP^" discussed the computation of Pisano's w„ for large n's. 
E. B. Escott"^ computed Sl/ti^. E. Landau^"' had evaluated Sl/wgA 
in terms of the sum of Lambert's^ series of Ch. X, and "Zl/uoh+i in relation 
to theta series. 
A. Tagiuri^^ employed the series Wi = l, U2 = l, Us = 2,... of Leonardo 
and the generalization Ui, U2,..., where t/„= t/„_i + i[7„_2, with Ui = a, 
U2 = b both arbitrary. Writing e for a^+ab — h^, it is proved that 
UJJ-Ur^.^U,^, = ( - 1)"- V^,+,_„e. 
{C/„4.a+( — l)*?7„_j)/?7„ is an integer independent of a, h, n; it equals 
Wj+i+Wj-i. It is shown that u^ is a multiple of u^ if and only if r is a 
multiple of s. 
Tagiuri^^ obtained analogous results for the series defined by Vn = hVn-i 
-\-lVn-2, and the particular series r„ obtained by taking ri = l, V2 = h. If 
h and I are relatively prime, v^ is a multiple of v^ if and only if r is a multiple 
of s. Let ^{i\) be the number of terms of the series of f's which are ^ r, and 
prime to it; if h>l, <i>(t',) is Euler's (f>{i); but, if h = l, <l>(r,) =0(^)+0(^y2), 
the last term being zero if i is odd. If i and j are relatively prime, $(y,y) 
=*(j'.)$(r;). 
Tagiuri^" proved that, for his series of v's, the terms between v^p and 
v^p+i) are incongruent modulo t'^. if h>l, and for /i = 1 except for Vkp+i=Vkp+2- 
If /x is not divisible by k and e is the least solution of /-**=! (mod f*), then 
Vj.= v^ (mod t';t) if a:=Ai (mod 4A:e). 
If /x is not divisible by k, and k is odd, and €1 is the least positive solution of 
P=l (mod Vk), then Vx=v^ (mod ft) if x=ij, (mod 2kei). 
A. Emmerich^^ proved that, in the series of Pisano, 
"Nature, 56, 1897, 10. «Math. Quest. Educat. Times, 3, 1903, 46; 4, 1903, 52 
""Math. Quest. Educat. Times, 66, 1897, 82-3; cf. 72, 1900, 40, 71. 
"*/6m/., 71, 1899,49-50. "«/6ui., Ill; 4, 1903, 107-8; 9, 1906, 55-7. 
«L'interm6diaire des math., 6, 1899, 76-7. 
*»Ibid., 7, 1900, 281, 312. "L'interm^diaire des math., 8, 1901, 63-64. 
•'"/Wd., 7, 1900, 172-7. ^'>>Ibid., 9, 1902, 43-4. 
•■<^BuU. Soc. Math. France, 27, 1899, 198-300. «'Periodico di Mat., 16, 1901, 1-12. 
"Peridico di Mat., 97-114. "/bid., 17, 1902, 77-88, 119-127. 
"Mathesis, (3), 1, 1901, 98-9. 
