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姓名 李冠葦(Guan-Wei Li)  查詢紙本館藏   畢業系所 數學系
論文名稱 關於方程式2x^2+1=3^n的研究
(The Diophantine Equation 2x^2+1=3^n)
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摘要(中) 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)。
摘要(英) 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).
關鍵字(中) ★ 方程式 關鍵字(英) ★ Binary Recurrent Sequence
★ Diophantine Equatoin
論文目次 1 Introduction
2 Preliminaries
3 The Equation 2x^2+1=3^n
References
參考文獻 1. F. Beukers, The Multiplicity of Binary Recurrences, Compositio
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.
指導教授 呂明光(Ming-Guang Leu) 審核日期 2002-6-7
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