On the Diophantine Equation of  (x^m-1)/(x-1)=(y^n-1)/(y-1)

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洪雅婷(Ya-Ting Hung) 查詢紙本館藏

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論文名稱

(On the Diophantine Equation of (x^m-1)/(x-1)=(y^n-1)/(y-1))

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★ On the Distribution of Primes

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摘要(中)

我們考慮特別的整係數方程式去尋找整數解或有理數解。Ratat和Goormaghtigh觀察出當x,y,m,n為正整數時，(x,y,m,n)=(5,2,3,5)和(90,2,3,13)是方程式 (x^m-1)/(x-1)=(y^n-1)/(y-1) 的解。因此，猜想此方程式只有這兩組解。現在，我們集中焦點在m=3。此時方程式有兩組已知的解。除了那兩組解之外的解就稱為例外解。這篇論文，主要是考慮當n=4時，此方程式沒有例外解。

摘要(英)

We consider special Diophantine equations with integral coefficient and seek
integral or rational solutions. Ratat[1] and Goormaghtigh [2] observed that
31=(2^5-1)/(2-1)=(5^3-1)/(5-1)
and 8191=(2^13-1)/(2-1)=(90^3-1)/(90-1)
are solutions of the Diophantine equation
(x^m-1)/(x-1)=(y^n-1)/(y-1)
; x > 1; y > 1; n > m > 2.....(1)
Now, we will focus our attention on the equation
(x^3-1)/(x-1)=(y^n-1)/(y-1)
; n > 2; x > 1; y > 1 with x > y.....(2)
Equation (2) has two known solutions (x, y, n) = (5, 2, 5), (90, 2, 13). Any other
solution (x, y, n) of (2) will be called exceptional. In this paper, we show that this
equation (2) has no exceptional solution when n = 4.

關鍵字(中)

關鍵字(英)

★ Diophantine Equation

論文目次

Contents
Abstract ........................................................................................ii
Outlines .......................................................................................iii
0 Introduction .................................................................................1
1 The Diophantine Equation (x^3-1)/(x-1)=(y^n-1)/(y-1).........................3
2 The Method of Descent .....................................................................6
3 The Diophantine Equation (x^3-1)/(x-1)=(y^4-1)/(y-1).....................12
References......................................................................................25

參考文獻

References
[1] R. Ratat L'intermediaire des Mathematiciens, 23 (1916),150.
[2] R. Goormaghtigh, L'intermediaire des Mathematiciens, 24 (1917), 88.
[3] A. Makowski, A. Schinzel, Sur I"equation indet'ermin'ee de R. Goormaghtigh, Mathesis 68 (1959), 128-142.
[4] A. Makowski, A. Schinzel, Sur I"equation indet'ermin'ee de R. Goormaghtigh, Mathesis 70 (1965), 94-96.
[5] T. Nagell, The diophantine equation x^2 + 7 = 2^n, Ark. Mat. 4 (1961) 185-187.
[6] S. Ramanujan, Question 464,J.Indian Math. Soc. 5 (1913), Collected Papers, Cambridge University Press. Cambridge, 1927, p. 327.
[7] T. N. Shorey, Some exponential diophantine equations, in: Number theory and related topics, Bombay (1989), 217-229.
[8] T. Nagell, Introduction to Number Theory, Wiley, New York, 1951.
[9] M.-H. Le, On the diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1),Trans. Amer. Math. Soc. 351 (1999), 1063-1074.
[10] G. Bergmann, U ber Eulers Beweis des grossen Fermatschen Satzes fur den Exponenten 3:, Math. Ann., 164(1996), 159-175.
[11] Maohua Le, Exceptional solutions of the exponential diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1), J. reine angew. Math. 543 (2002), 187-192.
[12] Yu. V. Nesterenko and T. N. Shorey, On an equation of Goormaghtigh, Acta Arith. 83 (1998), 381-389.
[13] Pingzhi Yuan, On the diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1), Journal of Number Theory 112 (2005), 20-25.
[14] Kenneth Ireland, Michael Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag New York Inc., 1982.
[15] Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer- Verlag New York Inc., 1986.
[16] L. J. Mordell, Diophantine Equations, Cambridge, England, 1969.25

指導教授

陳燕美(Yen-Mei J. Chen)

審核日期

2006-6-21

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