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研究生:李冠葦
研究生(外文):Guan-Wei Li
論文名稱:關於方程式2x^2+1=3^n的研究
論文名稱(外文):The Diophantine Equation 2x^2+1=3^n
指導教授:呂明光
指導教授(外文):Ming-Guang Leu
學位類別:碩士
校院名稱:國立中央大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:22
中文關鍵詞:方程式
外文關鍵詞:Binary Recurrent SequenceDiophantine Equatoin
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約在1913年左右 Ramanujan 猜想方程式x^2+7=2^n只有五組正整數解,此猜想在1948年由Nagell首先給出證明,其後有釵h數學家以不同的方法再一次證明此猜想。在1980年以後,釵h數學家開始研究方程式D_1x^2+D_2=λ^2k^n 的正整數解。在余茂華一系列的論文中證明除了方程式x^2+7=2^(n+2)有五組正整數解及方程式
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)。
Ramanujan observed in 1913 that
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
about equation (1.3).
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).
1 Introduction
2 Preliminaries
3 The Equation 2x^2+1=3^n
References
1. F. Beukers, The Multiplicity of Binary Recurrences, CompositioMath. 40 (1980), no. 2, 251-267.2. Y. Bugeaud and T. N. Shorey, On the number of solutions of thegeneralized Ramanujan-Nagell equation, J. reine angew. Math.539 (2001), 55-74.3. E. L. Cohen, The Diophantine Equation x^2 +11 = 3^k and RelatedQuestions, 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 ofimaginary quadratic fields, Kexue Tongbao 32 (1987), no. 10,724—727.(in Chinese)7. Maohua Le, On the number of solutions of the diophantine equa-tionx^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 GeneralizedRamanujan-Nagell Equation D_1x^2 + D_2 = 4p^n , J. NumberTh. 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 ofNumbers, 4th ed., John Wiley and Sons, 1980.15. S. Ramanujan, Collected Papers, Chelsea Publishing Co., NewYork, 1962, 327.16. H. S. Shapiro and D. L. Slotnick, On the mathematical theoryof error-correcting codes, IBM J. Res. Develop. 3 (1959), 25-34.17. Th. Skolem, S. Chowla and D. J. Lewis, The Diophantine equation2^(n+2)− 7 = x^2 and related problems, Proc. Amer. Math.Soc. 10 (1959), 663—669.
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