CHAP, xvii] RECUKRING SERIES; LUCAS' un, tv 395
common with Bm other than a divisor of n. If.p is an odd divisor of Bm and if h is the least k for which Bk is divisible by p, then h is a divisor of m. If p is an odd prime, Bp-\ or B^i is divisible by p according as b is a quadratic residue or non-residue of p, whatever be the value of a. This is used to prove the existence of prunes 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^ 1 [Ch. XVIII]. E. Lucas16 stated without proof theorems on the series of Pisano.1 The sum of the first n terms equals (7n+2— 2; the sum of those terms taken with alternate signs equals ( — l)"Un^i. Also
We have the symbolic formulas
Un(U-l)p,
where, after expansion, exponents are replaced by subscripts. Prom E. Catalan's Manuel des Candidats a Pficole Polytechnique, I, 1857, 86, he quoted
.5(n\-L*2(n\+ V sW + 5W + "T
Lucas17 employed the roots a, b of x2=o;+l and set
The w's form the series of Pisano with the terms 0, 1 prefixed, so that UQ= 0, Wi= ui== 1, w3 = 2. Since §un2—vn2= ±4, un and vn have no common factor other than 2. If p is a prime 5**2; 5, we have upz==±l, vp=l (mod p). We have the symbolic formulas
^^ = ^{^"(-1)*}^ (-l)k*un.kp=un{vk~uk}*.
Given a law Un+k=AQUn+P-}- . . . +APUn of recurrence, we can replace the symbol Uk by 0(J7), where
since C/n+Jbp= (7n{<^(C7)}p, symbolically.
E. Lucas18 stated theorems on the series of Pisano. We have
and his16 symbolic formulas with u's in place of U's. um is divisible by up and ug, and by their product if p, q are relatively prime. Set vn=uZn/un. Then
18Nouv. Correep. Math., 2, 1876, 74r-5.
"/bid., 201-6.
18Compte8 Rendus Paris, 82, 1876, 165-7.