On the diophantine equation $(x^m + 1)(x^n + 1) = y²$

• Autorzy: Maohua Le
• Języki publikacji: EN

Abstrakty

EN

1. Introduction. Let ℤ, ℕ, ℚ be the sets of integers, positive integers and rational numbers respectively. In [7], Ribenboim proved that the equation
   (1) $(x^m + 1)(x^n + 1) = y²$, x,y,m,n ∈ ℕ, x > 1, n > m ≥ 1,
has no solution (x,y,m,n) with 2|x and (1) has only finitely many solutions (x,y,m,n) with 2∤x. Moreover, all solutions of (1) with 2∤x satisfy max(x,m,n) < C, where C is an effectively computable constant. In this paper we completely determine all solutions of (1) as follows.
  Theorem. Equation (1) has only the solution (x,y,m,n)=(7,20,1,2).

Kategorie tematyczne

• 11D61: Exponential equations
• 11J86: Linear forms in logarithms; Baker's method

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1997
• Tom: 82
• Numer: 1
• Strony: 17-26

Daty

wydano

1997

otrzymano

1996-02-19

poprawiono

1996-09-19

Twórcy

autor

Maohua Le

• Department of Mathematics, Zhanjiang Teachers College, 524048 Zhanjiang, Guangdong, P.R. China

Bibliografia

• [1] C. Ko, On the diophantine equation $x² = y^n + 1$, xy ≠ 0, Sci. Sinica 14 (1964), 457-460.
• [2] M. Laurent, M. Mignotte et Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285-321.
• [3] M.-H. Le, A note on the diophantine equation x²p - Dy² = 1, Proc. Amer. Math. Soc. 107 (1989), 27-34.
• [4] W. Ljunggren, Zur Theorie der Gleichung x²+1 = Dy⁴, Avh. Norske Vid. Akad. Oslo I 5 (1942), no. 5, 27 pp.
• [5] W. Ljunggren, Sätze über unbestimmte Gleichungen, Skr. Norske Vid. Akad. Oslo I (1942), no. 9, 53 pp.
• [6] W. Ljunggren, Noen setninger om ubestemte likninger av formen $(x^n-1)/(x-1) = y^q$, Norsk. Mat. Tidsskr. 25 (1943), 17-20.
• [7] P. Ribenboim, Square classes of $(a^n-1)/(a-1)$ and $a^n+1$, Sichuan Daxue Xuebao, Special Issue, 26 (1989), 196-199.
• [8] N. Robbins, On Pell numbers of the form px², where p is a prime, Fibonacci Quart. 22 (1984), 340-348.
• [9] A. Rotkiewicz, Applications of Jacobi's symbol to Lehmer's numbers, Acta Arith. 42 (1983), 163-187.