# On the diophantine equation $\left({x}^{m}+1\right)\left({x}^{n}+1\right)=y²$

Acta Arithmetica (1997)

• Volume: 82, Issue: 1, page 17-26
• ISSN: 0065-1036

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## Abstract

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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) $\left({x}^{m}+1\right)\left({x}^{n}+1\right)=y²$, 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).

## How to cite

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Maohua Le. "On the diophantine equation $(x^m + 1)(x^n + 1) = y²$." Acta Arithmetica 82.1 (1997): 17-26. <http://eudml.org/doc/207074>.

@article{MaohuaLe1997,
abstract = {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).},
author = {Maohua Le},
journal = {Acta Arithmetica},
keywords = {exponential diophantine equations; Pell numbers; linear forms in two logarithms},
language = {eng},
number = {1},
pages = {17-26},
title = {On the diophantine equation $(x^m + 1)(x^n + 1) = y²$},
url = {http://eudml.org/doc/207074},
volume = {82},
year = {1997},
}

TY - JOUR
AU - Maohua Le
TI - On the diophantine equation $(x^m + 1)(x^n + 1) = y²$
JO - Acta Arithmetica
PY - 1997
VL - 82
IS - 1
SP - 17
EP - 26
AB - 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).
LA - eng
KW - exponential diophantine equations; Pell numbers; linear forms in two logarithms
UR - http://eudml.org/doc/207074
ER -

## References

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1. [1] C. Ko, On the diophantine equation $x²={y}^{n}+1$, xy ≠ 0, Sci. Sinica 14 (1964), 457-460.
2. [2] M. Laurent, M. Mignotte et Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285-321. Zbl0843.11036
3. [3] M.-H. Le, A note on the diophantine equation x²p - Dy² = 1, Proc. Amer. Math. Soc. 107 (1989), 27-34.
4. [4] W. Ljunggren, Zur Theorie der Gleichung x²+1 = Dy⁴, Avh. Norske Vid. Akad. Oslo I 5 (1942), no. 5, 27 pp. Zbl0027.01103
5. [5] W. Ljunggren, Sätze über unbestimmte Gleichungen, Skr. Norske Vid. Akad. Oslo I (1942), no. 9, 53 pp.
6. [6] W. Ljunggren, Noen setninger om ubestemte likninger av formen $\left({x}^{n}-1\right)/\left(x-1\right)={y}^{q}$, Norsk. Mat. Tidsskr. 25 (1943), 17-20.
7. [7] P. Ribenboim, Square classes of $\left({a}^{n}-1\right)/\left(a-1\right)$ and ${a}^{n}+1$, Sichuan Daxue Xuebao, Special Issue, 26 (1989), 196-199. Zbl0709.11014
8. [8] N. Robbins, On Pell numbers of the form px², where p is a prime, Fibonacci Quart. 22 (1984), 340-348.
9. [9] A. Rotkiewicz, Applications of Jacobi's symbol to Lehmer's numbers, Acta Arith. 42 (1983), 163-187. Zbl0519.10004

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