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1: 24.15 Related Sequences of Numbers
§24.15(iv) Fibonacci and Lucas Numbers
The Lucas numbers are defined by v 0 = 2 , v 1 = 1 , and v n + 1 = v n + v n - 1 , n 1 .
24.15.11 k = 0 n / 2 ( n 2 k ) ( 5 9 ) k B 2 k u n - 2 k = n 6 v n - 1 + n 3 n v 2 n - 2 ,
24.15.12 k = 0 n / 2 ( n 2 k ) ( 5 4 ) k E 2 k v n - 2 k = 1 2 n - 1 .
2: 27.22 Software
  • Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p - 1 , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

    For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

  • 3: 24.1 Special Notation
    The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
    4: Bibliography L
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • O. Lehto and K. I. Virtanen (1973) Quasiconformal Mappings in the Plane. 2nd edition, Springer-Verlag, New York.
  • É. Lucas (1891) Théorie des nombres. Tome I: Le calcul des nombres entiers, le calcul des nombres rationnels, la divisibilité arithmétique. Gauthier-Villars, Paris (French).
  • S. K. Lucas and H. A. Stone (1995) Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 217–231.
  • S. K. Lucas (1995) Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.
  • 5: Bibliography R
  • H. Rademacher (1973) Topics in Analytic Number Theory. Springer-Verlag, New York.
  • S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,
  • J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
  • K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.