| Diophantine方程组a~2+b~2=c~r和a~x+b~y=c~z的一点注记 |
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| 引用本文： | 乐茂华. Diophantine方程组a~2+b~2=c~r和a~x+b~y=c~z的一点注记[J]. 数学学报, 2008, 51(4): 677-684. DOI: CNKI:SUN:SXXB.0.2008-04-008 |
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| 作者姓名： | 乐茂华 |
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| 作者单位： | 湛江师范学院数学系 湛江 524048 |
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| 摘 要： | 设r是大于1的正奇数,m是正偶数,V(r)+U(r)(-1)~(1/2)=(m+(-1)~(1/2))~r.本文证明了:当a=|V(r)|,b=|U(r)|,c=m~2+1时,如果r≡5(mod8),m>r~2且r<11500或者m>2r/π且r>11500,则方程a~x+b~y=c~z仅有正整数解(x,y,z)=(2,2,r).
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| 关 键 词： | 指数Diophantine方程 广义Fermat猜想 二次剩余 |
| 收稿时间： | 2006-03-08 |
A Note on the Diophantine System a~2+b~2=c~r and a~x+b~y=c~2 |
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| Affiliation: | Mao Hua LE Department of Mathematics,Zhanjiang Normal College,Zhanjiang 524048,P.R.China |
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| Abstract: | Let r be a positive odd integer with r>1,and let m be a positive even integer.Further move let a=|V(r)|,b=|U(r)| and c=m~2+1,where V(r)+ U(r)(-1)~(1/2)=(m+(-1)~(1/2))~r.In this paper we prove that if r≡5 (mod 8) and either m>r~2,r<11500 or m>2r/π,r>11500,then the equation a~x+b~y=c~z has only the positive integer solution (x,y,z)=(2,2,r). |
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| Keywords: | exponential Diophantine equation generalized Fermat conjecture quadratic residue |
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