# On the diophantine equation $({x}^{m}+1)({x}^{n}+1)=y\xb2$

Acta Arithmetica (1997)

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

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topMaohua 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

top- [1] C. Ko, On the diophantine equation $x\xb2={y}^{n}+1$, 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. Zbl0843.11036
- [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. Zbl0027.01103
- [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. Zbl0709.11014
- [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. Zbl0519.10004

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