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Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation

Prapanpong Pongsriiam
대한수학회 (The Korean Mathematical Society)|2017.07|대한수학회논문집 (Communications of the KMS)|제32권 제3호|pp. 511-522

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키워드

Fibonacci numberDiophantine equationLucas numberBrocard-Ramanujan equation
주제분류수학

초록

We explicitly solve the diophantine equations of the form $$ A_{n_1}A_{n_2}\cdots A_{n_k}\pm 1 = B_m^2, $$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques \cite{Mar} and L. Szalay \cite{Sza} concerning a variant of Brocard-Ramanujan equation.

Abstract

We explicitly solve the diophantine equations of the form $$ A_{n_1}A_{n_2}\cdots A_{n_k}\pm 1 = B_m^2, $$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques \cite{Mar} and L. Szalay \cite{Sza} concerning a variant of Brocard-Ramanujan equation.

참고문헌 (14)

  1. B. C. Berndt, "On the Brocard–Ramanujan Diophantine equation n! + 1 = m2", Ramanujan J, 4(1), pp.41~42 (2006)
  2. Y. Bilu, "Existence of primitive divisors of Lucas and Lehmer numbers", J. Reine Angew. Math, 539, pp.75~122 (2001)
  3. Brother A. Brousseau, "Fibonacci and Related Number Theoretic Tables" (1972), The Fibonacci Association
  4. R. D. Carmichael, "On the numerical factors of the arithmetic forms αn ± βn", Ann. of Math.(2), 15, pp.30~48 (1913)
  5. A. D¸abrowski, "On the Brocard–Ramanujan problem and generalizations", Colloq. Math, 126(1), pp.105~110 (2012)
  6. A. D¸abrowski, "Variations on the Brocard–Ramanujan equation", J. Number Theory, 133(4), pp.1168~1185 (2013)
  7. Y. Kunrui, "p-adic logarithmic forms and a problem of Erd¨os", Acta Math, 211(2), pp.315~382 (2013)
  8. F. Luca, "The Diophantine equation P(x) = n! and a result of M. Overholt", Glas. Mat. Ser. III, 37(2), pp.269~273 (2002)
  9. D. Marques, "The Fibonacci version of the Brocard–Ramanujan Diophantine equation", Port. Math, 68(2), pp.185~189 (2011)
  10. P. Pongsriiam, "Fibonacci and Lucas numbers which are one away from their products", ~
  11. P. Pongsriiam, "Factorization of Fibonacci numbers into products of Lucas numbers and related results", JP J. Algebra Number Theory Appl, 38(4), pp.363~372 (2016)
  12. C. L. Stewart, "On divisors of Lucas and Lehmer numbers", Acta Math, 211(2), pp.291~314 (2013)
  13. L. Szalay, "Diophantine equations with binary recurrences associated to the Brocard–Ramanujan problem", Port. Math, 69(3), pp.213~220 (2012)
  14. M.Ward, "The prime divisors of Fibonacci numbers", Pacific J. Math, 11, pp.379~386 (1961)

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