On the diophantine equation (xm+1)(xn+1)=y²
Maohua Le
Acta Arithmetica, Tome 80 (1997), p. 17-26 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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) (xm+1)(xn+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).

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:207074 
@article{bwmeta1.element.bwnjournal-article-aav82i1p17bwm,
     author = {Maohua Le},
     title = {On the diophantine equation $(x^m + 1)(x^n + 1) = y$^2$$
            },
     journal = {Acta Arithmetica},
     volume = {80},
     year = {1997},
     pages = {17-26},
     zbl = {0893.11013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav82i1p17bwm}
}

Maohua Le. On the diophantine equation $(x^m + 1)(x^n + 1) = y²$
            . Acta Arithmetica, Tome 80 (1997) pp. 17-26. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav82i1p17bwm/

[00000] [1] C. Ko, On the diophantine equation x²=yn+1, xy ≠ 0, Sci. Sinica 14 (1964), 457-460.

[00001] [2] M. Laurent, M. Mignotte et Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285-321. | Zbl 0843.11036

[00002] [3] M.-H. Le, A note on the diophantine equation x²p - Dy² = 1, Proc. Amer. Math. Soc. 107 (1989), 27-34.

[00003] [4] W. Ljunggren, Zur Theorie der Gleichung x²+1 = Dy⁴, Avh. Norske Vid. Akad. Oslo I 5 (1942), no. 5, 27 pp. | Zbl 0027.01103

[00004] [5] W. Ljunggren, Sätze über unbestimmte Gleichungen, Skr. Norske Vid. Akad. Oslo I (1942), no. 9, 53 pp.

[00005] [6] W. Ljunggren, Noen setninger om ubestemte likninger av formen (xn-1)/(x-1)=yq, Norsk. Mat. Tidsskr. 25 (1943), 17-20.

[00006] [7] P. Ribenboim, Square classes of (an-1)/(a-1) and an+1, Sichuan Daxue Xuebao, Special Issue, 26 (1989), 196-199. | Zbl 0709.11014

[00007] [8] N. Robbins, On Pell numbers of the form px², where p is a prime, Fibonacci Quart. 22 (1984), 340-348.

[00008] [9] A. Rotkiewicz, Applications of Jacobi's symbol to Lehmer's numbers, Acta Arith. 42 (1983), 163-187. | Zbl 0519.10004