关于Diophantine方程a^x+b^y=c^z的Terai猜想

On Terai's Conjecture Concerning the Diophantine Equation a^x+b^y=c^z

设m是正偶数.证明了(A)若b是奇素数,且a=m|m^6-21m^4+35m^2-7|,b=|7m^6-35m^4+21m^2-1|,c=m^2+1,则Diophantine方程G:a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,7);(B)若m>2863,且a=m|m^8-36m^6+126m^4-84m^2+9|,b=|9m^8-84m^6+126m^4-36m^2+1|,c=m^2+1,则Diophantine方程G仅有正整数解(x,y,z)=(2,2,9);(C)若a,b,c适合a=m|∑_(i=0)^((r-1)/2)(-1)~i(_(2i)~r)m^(r-2i-1)|,b=|∑_(i=0)^((r-1)/2)(-1)~i(_(2i+1)~r)m^(r-2i-1)|,c=m^2+1,r≡1(mod4),2|x,2|y,且b为奇素数或m>145r(log r),则方程G仅有解(x,y,z)=(2,2,r).

Let m be an even positive integer. In this paper, we prove that （A） if b is an odd prime and a = m ｜m^6- 21m^4 ＋ 35m^2- 7｜, b = ｜7m^6- 35m^4 ＋ 21m^2-1｜, c = m^2＋1, then the Diophantine equation G： a^x ＋ b^y = c^z has only one positive integer solution （x, y, z）=（2, 2, 7）; （B） ifm 〉 2863, and a = m｜m^8-36m^6＋126m^4-84m^2＋9｜, b = ｜9m^8-84m^6＋126m^4-36m^2＋ 1｜, c = m^2 ＋ 1, then the Diophantine equation G has only one positive integer solution （x, y, z） =（2,2,9）; （C） if a,b,c satisfy a=m｜∑r-1/2 i=0（-1）^i（2i^r）m^r-2i-1｜,b=｜∑r-1/2 i=0（-1）^i（2i＋1^r）m^r-2i-1｜,c=m^2＋1,r≡1（mod4）,2｜x,2｜y, and b is an odd prime or m 〉 145r（logr）, then the Diophantine equation G has only one positive integer solution （x, y, z） = （2, 2, r）.