The Exponential Diophantine Equations 1+p^a+q^b+r^c=s^d

• Main Authors: SUH-JI YEH, 葉素姬
• Other Authors: YU-CHING HUNG
• Format: Others
• Language: zh-TW
• Published: 1993
• Online Access: http://ndltd.ncl.edu.tw/handle/78193879998393873834

碩士 === 國立師範大學 === 數學系 === 81 === Exponential Diophantine 方程式是指方程式Σ±x(i)=0，其中限制Πx( i)為特定的質數所整除，例如1+2^a+7^b=5^c、1+2^a+2^b*7^c =7^d都是 ；早期對Diophantine 方程的研究可在數論史上發現，近期的成果則起源 於對有限群特徵理論(character theory)的研究。解出Diophantine 方程 式的所有解，得以研究某些有限群的特性，例如在[2]這篇論文，作者利 用方程式1+2^a=3^b*5^c+2^d*3^e*5^f的解研究一些單群的特性，在[3]、 [4]這二篇論文都有Diophantine方程的應用；然而到目前為止仍有一些 Diophantine方程未被解出，例如方程式1+2^a*3^b=5^c+2^d*3^e*5^f，當 c=f=0,a=d,b=e 時有無限多個顯然的解，但仍不知道是否有有限個非顯然 的解，在[5]更有詳細列出一些未被解出的Diophantine 方程。在這篇 論文我們解出二種形式的Diophantine 方程，二、三節是解出形式 1+x+ y=z 的方程式，這是接續[1]的工作，第四節我們求出 1+p^a+q^b+r^c= s^d 的所有解，其中{p,q,r,s}={2,3,5,7}，基本上都是用"同餘"的方法 解Diophantine 方程，這篇論文的作法是限制方程式右邊數小於20的情況 下，先用電腦跑出其解，而後我們猜測右邊數的指數大於20的情況下不再 有解，試圖用反證法證明之。 The equations which have the form Σ±x(i)=0 where the primes dividing Πx(i) are specified are called Exponential Diophantine equations﹒For example，the equations 1+2^a+7^b =5^c and 1+2^a+2^b*7^c=7^d are both Exponential Diophantine equations﹒The investigations of Exponential Diophantine equations occur very early in the history of the theory of numbers﹒The recent results arise naturally in the character theory of finite groups﹒ If one finds all solutions of Exponential Diophantine equations，he can further investigate the characteristics of some finite simple groups﹒For example， the authors in [2] have used the solutions to the equation 1+2^a=3^b*5^c +2^d*3^e*5^f to characterize some simple groups﹒ In [3]、[4] there are applications of Exponential Diophantine equations﹒ But there are also a few Diophantine equations which have not been solved till now﹒For example，the still- unsolved equation 1+2^a*3^b=5^c+2^d*3^e*5^f has infinitely many solutions of the form c=f=0,a=d,b=e﹒It is unknown whether such equations must have only a finite number of nontrivial solutions ﹒ In this paper we have solved two forms of Diophantine equations﹒In Section Two and Three the form 1+x+y=z where x、y and z are positive integers divided by 2,3,5,7 is solved﹒ The part continues the second section of [1]﹒In Section Four we find all solutions of the equation 1+p^a+q^b+r^c=s^d where {p,q, r,s}={2,3,5,7}﹒In general the techniques which solve such equations involve careful consideration of the equation modulo a series of prime , prime power and certain other moduli﹒We use the computer to run out the solutions by limitting the exponents of the right-side numbers of the equations smaller than 20﹒Then we conjecture that there is no more solution when the exponents of the right-side number are bigger than 20﹒And we try to prove them﹒