Plus petit commun multiple des termes consécutifs d'une suite récurrente linéaire

• Autores: Jean Paul Bézivin
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 40, Fasc. 1, 1989, págs. 1-11
• Idioma: francés
• Títulos paralelos:
  • Mínimo común múltiplo de los términos consecutivos de una sucesión recurrente lineal
  • Least common multiple of consecutive terms of a linear recurrent sequence
• Enlaces
  • Texto completo
• Resumen

  • Let $a,b$ be coprime rational integers, and $u(n)$ the binary recurrent sequence $u(n+2)=au(n+1)+bu(n)$, with the initial values $u(0)=0$ and $u(1)=1$. It is proved in [4] that the quotient of the logarithm of the product of the $u(k), 1\leq k\leq n$, and of the logarithm of the least common multiple of the $ u(k), 1\leq k\leq n$, converges to $\pi^2/6$ when $n$ goes to infinity. In this paper, we generalize this result to other recurrent sequences. As an example, fot the binary sequence above with initial values $u(0)=2$ and $u(1)=a$, the limit is $\pi^2/8$.

• Referencias bibliográficas
  • T. Apostol, Resultants of cyclotomic polynomials, Proc. Amer. Math. Soc. 24 (1970), 457–462.
  • R. Ayoub, An Introduction to the Analytic Number Theory, AMS Mathematical Surveys 10, American Mathematical Society, Providence, 1963.
  • M. Davis, Hilbert’s tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233–269.
  • P. Kiss and F. Matyas, An asymptotic formula for $\pi$, J. Number Theory 31 (1989), 255–259.
  • Y. Matiyasevitch and R. Guy, A new formula for $\pi$, Amer. Math. Monthly 93 (1986), 631–635.
  • A. Schinzel, Primitive divisors of $A^n - B^n$, J. Reine Angew. Math. 268–269 (1974), 27–33.