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CHAPTER XVII. RECURRING SERIES; LUCAS* «„, t». Leonardo Pisano1, 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 pan- 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 Girard2 noted the law un+2=un+i-\-un for these series. Robert Simson3 noted that this series is given by the successive conver-gents to the continued fraction for (V5+l)/2. The square of any term is proved to differ from the product of the two adjacent terms by =*=!. L. Euler4 noted that (a+Vb)k=Ak+Bk<y/b implies J. L. Lagrange5 noted that the residues of Ak and Bk with respect to any modulus are periodic. Lagrange6 proved that if the prime p divides no number of the form f—au2, then p divides a number of the form { (t+u Va)p+1 - («- uVi)p+1}/ Va. A. M. Legendre7 proved that, if 02-Ai^2 = l, then (0+^VZ)fl— 1 is of the form r+s\/A.y where r and s are divisible by a prime o>, not dividing C. F. Gauss8 proved [Lagrange' s6 result] that, if 6 is a quadratic non-residue of the prime p, then Bp+1 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 Bp+i. G. L. Dirichlet9 proved that, if b is an integer not a square and x is any integer prime to 6, and if U, V are polynomials in x, b such that 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 2ran+l. No prime of which b is a quadratic non-residue is a factor of V unless it be of the form 2mn — l. Lagrange6 had proved conversely that a 'Scritti, I, 1857 (Liber Abbaci), 283-4. 2L'Arithme*tique de Simon Stevin de Bruges, par Albert Girard, Leyde, 1634, p. 677. Lea Oeuvres Math, de Simon Stevin, 1634, p. 169. •Phil. Trans. Roy. Soc. London, 48, 1, 1753, 368-376; abridged edition, 10, 1809, 430-4. 4Novi 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. M6m. Ac. Berlin, ann^e 1775 (1777), 343; Oeuvres, 3, 782-3. 7The*orie des nombres, 1798, p. 457; ed. 2, 1808, p. 429; ed. 3, 1830, vol. 2, Art. 443, pp. 111-2. 8Disq. Arith., 1801, Art. 123. 'Deformislinearibus, Breslau, 1827; Werke, 1, 51. Cf. Kronecker." 393