Chap. XVII] Recueeing Seeies; Lucas' Un, v^. 395 
common with B^ other than a divisor of n. If p is an odd divisor of B^^ and 
if h is the least k for which B^ is divisible by p, then /i is a divisor of m. If p 
is an odd prime, Bp_i or Bp^i is divisible by p according as 6 is a quadratic 
residue or non-residue of p, whatever be the value of a. This is used to 
prove the existence of primes of the two forms n'2;±l(n a prime >2) and 
the existence of an infinitude of primes of each of the forms mz^l [Ch . XVIII] . 
E. Lucas^^ stated without proof theorems on the series of Pisano.^ The 
sum of the first n terms equals C/„+2 — 2; the sum of those terms taken with 
alternate signs equals ( — l)"C/„_i. Also 
U „_i -\-U n— U2n, UnUn+l ~ U n—\ U n—2 = U 2n} U n'^U n+1 ~ U „_i = C/ 3„+2' 
We have the symbolic formulas 
where, after expansion, exponents are replaced by subscripts. From 
E. Catalan's Manuel des Candidats a I'Ecole Polytechnique, I, 1857, 86, he 
quoted 
Lucas^'^ employed the roots a,boix^ = x-\-l and set 
^ ^ n \ 1.71 ^2re I 
a — Un 
The u's form the series of Pisano with the terms 0, 1 prefixed, so that 
Uo=0, Ui = U2=l, U3 = 2. Since 5w„^ — z^„^ = ± 4, u^ and Vn have no common 
factor other than 2. If p is a prime ^2, 5, we have Up=±l, Vp = l (mod 
p). We have the symbolic formulas 
Given a law Un+k = -^oUn+p+ ■ ■ ■ -i-ApUn of recurrence, we can replace the 
symbol f/* by ( C/) , where 
(f){u)=Aou''+Aiu''-^+. . . +Ap_^u+A 
p) 
since U^+kp^ U''{({){U)}^, symbolically. 
E. Lucas^^ stated theorems on the series of Pisano. We have 
2"\/5i/, = (l + V5r-(l-V5r, ^n+i = l + (i) + (''2^) + --- 
and his^^ symbolic formulas with u's in place of U^s. Up^ is divisible by Up 
and Uq, and by their product if p, q are relatively prime. Set Vn = U2n/un. 
Then 
Vn+2 = Vn+l+Vn, Vin = An-2, y4„+2 = «^Wl +2- 
«Nouv. Corresp. Math., 2, 1876, 74r-5. 
^Ubid., 201-6. 
"Comptes Rendus Paris, 82, 1876, 165-7. 
