# Lucas numbers

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##### 1: 24.15 Related Sequences of Numbers
###### §24.15(iv) Fibonacci and LucasNumbers
The Lucas numbers are defined by $v_{0}=2$, $v_{1}=1$, and $v_{n+1}=v_{n}+v_{n-1}$, $n\geq 1$.
24.15.11 $\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}\left(\frac{5}{% 9}\right)^{k}B_{2k}u_{n-2k}=\frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2},$
24.15.12 $\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}\left(\frac{5}{% 4}\right)^{k}E_{2k}v_{n-2k}=\frac{1}{2^{n-1}}.$
##### 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 $E_{n}$, $E_{n}\left(x\right)$, 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.