### 博碩士論文 89221014 詳細資訊

 以作者查詢圖書館館藏 、以作者查詢臺灣博碩士 、以作者查詢全國書目 、勘誤回報 、線上人數：4 、訪客IP：3.239.95.36

(The Diophantine Equation 2x^2+1=3^n)

 ★ 關於胡-黃-王猜測的研究 ★ 關於 (2,n) 群試問題的研究 ★ 關於方程式 x^2+11=p^n 的研究 ★ On problems of certain arithmetic functions ★ 伽羅瓦理論 ★ k階歐幾里得環 ★ On a Paper of P. M. Cohn ★ On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic ★ On Generalized Euclidean Rings ★ ZCm 的理想環生成元個數之上限 ★ Linearly Independent Sets and Transcendental Numbers

1. 本電子論文使用權限為同意立即開放。
2. 已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
3. 請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

3x^2+5=2^(n+2)、x^2+11=2^2×3^n、x^2+19=2^2×5^n各有三組正整數解外，方程式D_1x^2+D_2=λ^2k^n 的正整數解個數最多二組。此篇論文主要討論方程式2x^2+1=3^n的正整數解，由余茂華之前的結果得到方程式2x^2+1=3^n的正整數解個數最多二組，而在Begeard 和 Sheory 的論文中給定此方程式的正整數解分別為(x,n)=(1,1)及(2,2)，但是除了n=1,2，我們發現n=5也是此方程式的一個解。在此篇論文中，我們將求正整數解的問題轉換成二次遞迴數列(binary recurrent sequence)的問題並且利用Beukers論文中的相關結果來證明方程式2x^2+1=3^n僅有三組正整數解(x,n)=(1,1)，(2,2)以及(11,5)。

1^2 + 7 = 2^3 , 3^2 + 7 = 2^4 , 5^2 + 7 = 2^5 ,
11^2 + 7 = 2^7 , (181)^2 + 7 = 2^15 .(1.1)
When looking for the solutions of the equation
x^2 + 7 = 2^n in integers x > 0, n > 0,(1.2)
Ramanujan conjectured in 1913 that all the solutions of equation
(1.2) are given by (1.1). This was proved by Nagell in 1948 and by others, using several different proofs. The equation (1.2) is called the
Ramanujan-Nagell equation and has applications to binary error-correcting codes.
After 1980’s, many mathematicians concentrated on studying the equation
D_1x^2 + D_2 =λ^2 k^n, (1.3)
We denote by N(λ, D_1, D_2, k) the number of solutions(x, n) of the equation (1.3). For D_1 = 1, equation (1.3) is usually
called the generalized Ramanujan-Nagell equation. In a series of papers, Le proved that N(λ,D_1, D_2,p)≦2 except for
N(2, 1, 7, 2) = 5 and N(2, 3, 5, 2) = N(2, 1, 11, 3) = N(2, 1, 19, 5) =3. Bugeaud and Shorey offer the newest results and related ref-erences
In this paper, we are looking for solutions of the equation 2x^2 + 1 = 3^n in integer x≧1, n≧1. (1.4)
The previous result of Le implies N(1, 2, 1, 3) ≦2, and Bugeaud and Shorey refer that equation (1.4) has two solutions which are
given by n = 1, 2. But except for n = 1, 2, we find n = 5 is also a solution of equation (1.4). In this paper we will reduce the problem
of determining N(1, 2, 1, 3) to a binary recurrent sequence and use the results of Beukers [1] to prove that the equation (1.4) has only
the solutions (x, n) = (1, 1), (2, 2) and (11, 5).

★ Diophantine Equatoin

2 Preliminaries
3 The Equation 2x^2+1=3^n
References

Math. 40 (1980), no. 2, 251-267.
2. Y. Bugeaud and T. N. Shorey, On the number of solutions of the
generalized Ramanujan-Nagell equation, J. reine angew. Math.
539 (2001), 55-74.
3. E. L. Cohen, The Diophantine Equation x^2 +11 = 3^k and Related
Questions, Math. Scand. 38 (1976), no. 2, 240-246.
4. T. W. Hungerford, Algebra, Springer-Verlag, 1974.
5. W. Johnson, The Diophantine Equation x^2 + 7 = 2^n , Amer.
Math. Monthly 94 (1987), no. 1, 59-62.
6. Maohua Le, The divisibility of the class number for a class of
imaginary quadratic fields, Kexue Tongbao 32 (1987), no. 10,
724–727.(in Chinese)
7. Maohua Le, On the number of solutions of the diophantine equa-tion
x^2 + D = p^n , C. R. Acad. Sci. paris S´er. A 317 (1993),
135-138.
8. Maohua Le, On the Diophantine Equation D_1x^2 + D_2 = 2^(n+2),
Acta Arith. 64 (1993), 29-41.
9. Maohua Le, A note on the Generalized Ramanujan-Nagell Equation,
J. Number Th. 50 (1995), 193-201.
10. Maohua Le, A Note on the Number of Solutions of the Generalized
Ramanujan-Nagell Equation D_1x^2 + D_2 = 4p^n , J. Number
Th. 62 (1997), 100-106.
11. Maohua Le, On the Diophantine Equation (x^3− 1)/(x − 1) =(y^n− 1)/(y − 1), Trans. Amer. Math. Soc. 351 (1999), 1063-1074.
12. D. A. Marcus, Number fields, Springer-Verlag, 1987.
13. T. Nagell, The Diophantine Equation x^2 + 7 = 2^n , Ark. Mat. 4
(1961), 185-187.
14. I. Niven, H. S. Zuckerman, An Introduction to the Theory of
Numbers, 4th ed., John Wiley and Sons, 1980.
15. S. Ramanujan, Collected Papers, Chelsea Publishing Co., New
York, 1962, 327.
16. H. S. Shapiro and D. L. Slotnick, On the mathematical theory
of error-correcting codes, IBM J. Res. Develop. 3 (1959), 25-34.
17. Th. Skolem, S. Chowla and D. J. Lewis, The Diophantine equation
2^(n+2)− 7 = x^2 and related problems, Proc. Amer. Math.
Soc. 10 (1959), 663–669.