关于指数Diophantine方程x~2=D~(2m)-D~mp~n+p~(2n)
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摘要
设D> 1是正整数,p是适合p?D的素数.本文研究了指数Diophantine方程x~2=D~(2m)-D~mp~n+p~(2n)的满足m> 1的正整数解.根据Diophantine方程的性质,结合已有的结论,运用初等方法确定了方程满足m> 1的所有正整数解(D,p,x,m,n).这个结果修正并完整解决了文献[4]的猜想.
Let D be a positive integer with D > 1, and p be a prime with p ? D. In this paper,we study the positive integer solutions of the Diophantine equation x~2 = D~(2m)-D~mp~n+ p~(2n) with m > 1. By using properties and several known results of Diophantine equations with some elementary methods, all positive integer solutions(D, P, x, m, n) of the equations x~2 = D~(2m)-D~mp~n+p~(2n)are determined, which corrects and completely solves the presumption in [4].
Let D be a positive integer with D > 1, and p be a prime with p ? D. In this paper,we study the positive integer solutions of the Diophantine equation x~2 = D~(2m)-D~mp~n+ p~(2n) with m > 1. By using properties and several known results of Diophantine equations with some elementary methods, all positive integer solutions(D, P, x, m, n) of the equations x~2 = D~(2m)-D~mp~n+p~(2n)are determined, which corrects and completely solves the presumption in [4].
引文
[1]陈景润.关于Jesmanowicz猜想[J].四川大学学报(自然科学版),1962,(2):19-25.
[2]乐茂华,胡永忠.广义Lebesgue-Ramanujan-Nagell方程研究的新进展[J].数学进展,2012,41(4):385-396.
[3]佟瑞洲.关于丢番图方程P~(2z)-P~zD~m+D~(2m)=X~2(I)[J].沈阳农业大学学报(自然科学版),2004,35(3):283-285.
[4]佟瑞洲.一个丢番图方程的求解[J].沈阳师范大学学报(自然科学版),2005, 23(2):133-136.
[5]华罗庚.数论导引[M].北京:科学出版社,1979.
[6] Birkhoff G D, Vandivier H S. On the integral divisors of a~n-b~n[J]. Ann. Math., 1904, 5(2):173-180.
[7]曹珍富.丢番图方程引论[M].哈尔滨:哈尔滨工业大学出版社,1989.
[8] Bennett M A, Skinner C M. Ternary Diophantine equations via Galois representations and modular forms[J]. Canad. J. Math., 2004, 56(1):23-54.
[9] Bugeaud Y, Mignotte M. On the Diophantine equation(x~n-1)/(x-1)=yq with nagative x[A].Bennet M A. Number theory for the millennium I[C]. MA:Peters A. K., 2002:145—151.
[10] Mihailescu P. Primary cyclotomic units and a proof of Catalan's conjecture[J]. J. Reine Angew.Math., 2004, 572:167-195.
[11] Darmon M, Merel L. Winding quotients and some variants of Fermat's last theorem[J]. J. Reine Angew. Math., 1997, 490:81-100.
[2]乐茂华,胡永忠.广义Lebesgue-Ramanujan-Nagell方程研究的新进展[J].数学进展,2012,41(4):385-396.
[3]佟瑞洲.关于丢番图方程P~(2z)-P~zD~m+D~(2m)=X~2(I)[J].沈阳农业大学学报(自然科学版),2004,35(3):283-285.
[4]佟瑞洲.一个丢番图方程的求解[J].沈阳师范大学学报(自然科学版),2005, 23(2):133-136.
[5]华罗庚.数论导引[M].北京:科学出版社,1979.
[6] Birkhoff G D, Vandivier H S. On the integral divisors of a~n-b~n[J]. Ann. Math., 1904, 5(2):173-180.
[7]曹珍富.丢番图方程引论[M].哈尔滨:哈尔滨工业大学出版社,1989.
[8] Bennett M A, Skinner C M. Ternary Diophantine equations via Galois representations and modular forms[J]. Canad. J. Math., 2004, 56(1):23-54.
[9] Bugeaud Y, Mignotte M. On the Diophantine equation(x~n-1)/(x-1)=yq with nagative x[A].Bennet M A. Number theory for the millennium I[C]. MA:Peters A. K., 2002:145—151.
[10] Mihailescu P. Primary cyclotomic units and a proof of Catalan's conjecture[J]. J. Reine Angew.Math., 2004, 572:167-195.
[11] Darmon M, Merel L. Winding quotients and some variants of Fermat's last theorem[J]. J. Reine Angew. Math., 1997, 490:81-100.