## On the exponential Diophantine equation $$a^x+b^y=c^z$$.(English)Zbl 1009.11026

Let $$a$$, $$b$$, $$c$$ be fixed positive integers such that $$\min(a,b,c)>1$$ and $$\gcd(a,b,c)=1$$. In this paper the author proves that if $$a,b,c$$ satisfy some congruence conditions, then the equation $$(*)$$ $$a^x+b^y=c^z$$ has only one positive integer solution $$(x,y,z)$$. For example, he proves that if $$a\equiv -1\pmod{b^2}$$, $$b$$ is an odd prime with $$b\equiv 3\pmod 4$$ and $$a^2+b=c$$, then $$(*)$$ has only the positive integer solution $$(x,y,z)= (2,1,1)$$.

### MSC:

 11D61 Exponential Diophantine equations

### Keywords:

congruence conditions
Full Text:

### References:

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