CHAPTER XVII. 
RECURRING SERIES; LUCAS* Un, Vn. 
Leonardo Pisano\ or Fibonacci, employed, in 1202 (revised manuscript, 
1228), the recurring series 1, 2, 3, 5, 8, 13, ... in a problem on the number of 
offspring of a pair of rabbits. We shall write Un for the nth term, and Un 
for the {n-\-l)th term of 0, 1, 1, 2, 3, 5, . . . derived by prefixing 0, 1 to the 
former series. 
Albert Girard^ noted the law w„+2 = ^<n+i+'Wn for these series. 
Robert Simson^ noted that this series is given by the successive conver- 
gents to the continued fraction for (\/5 + l)/2. The square of any term is 
proved to differ from the product of the two adjacent terms by =*= 1. 
L. Euler^ noted that {a-\-\/b)'' = Ak+Bk^ implies 
A, = h{{a-\-Vbr+ia-Vbr}, B, = -~^{ia+Vbr-{a-Vby]. 
J. L. Lagrange^ noted that the residues of Ak and B^ with respect to any 
modulus are periodic. 
Lagrange^ proved that if the prime p divides no number of the form 
f—au^, then p divides a number of the form 
{{t-huVar+'-it-uV^)^']/Va. 
A. M. Legendre' proved that, if (f)^—A^^ = l, then ((l>+\l/VAy — l is 
of the form r+sV^? where r and s are divisible by a prime w, not dividing 
8=<o-lit(i) = +l, «=«+litg) = -l. 
C. F. Gauss* proved [Lagrange's^ result] that, if 6 is a quadratic non- 
residue of the prime p, then Bp+i is divisible by p for every integral value 
of a. If e is a divisor of p+ 1, then Be is divisible by p for e — 1 values of a, 
being a factor of B^+i. 
G. L. Dirichlet^ proved that, if b is an integer not a square and x is any 
integer prime to b, and if U, V are polynomials in x, b such that 
{x+Vbr=^u-\-vvb, 
then U and V have no common odd divisors. If n is an odd prime, no prime 
of which 6 is a quadratic residue is a factor of V unless it be of the form 
2mn+l. No prime of which 6 is a quadratic non-residue is a factor of V 
unless it be of the form 2mn — l. Lagrange^ had proved conversely that a 
iScritti, I, 1857 (Liber Abbaci), 283-4. 
^L'Arithm^tique de Simon Stevin de Bruges, par Albert Girard, Leyde, 1634, p. 677. Lea 
Oeuvres Math, de Simon Stevin, 1634, p. 169. 
3Phil. Trans. Roy. Soc. London, 48, I, 1753, 368-376; abridged edition, 10, 1809, 430-4. 
*Novi Comm. Acad. Petrop., 18, 1773, 185; Comm. Arith., 1, 554. 
^Additions to Euler's Algebra, 2, 1774, §§ 78-9, pp. 599-607. Euler, Opera Omnia, (1), 1, 619. 
•Nouv. M^m. Ac. Berlin, ann^e 1775 (1777), 343; Oeuvres, 3, 782-3. 
'TWorie des nombres, 1798, p. 457; ed. 2, 1808, p. 429; ed. 3, 1830, vol. 2, Art. 443, pp. 111-2. 
«Disq. Arith., 1801, Art. 123. »Deformishnearibus,Breslau,1827; Werke, 1,51. Cf . Kronecker." 
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