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Pillai's conjecture is that for each $k$, there are only finitely many solutions.

The ABC conjecture implies Pillai's conjecture as follows. First, I state a form of the ABC conjecture. Given three relatively prime positive integers $A+B=C$, the quality of the triple $(A,B,C)$ is $\log(C)/\log(R)$, where $R$ is the product of the primes that divide $ABC$. For example, the quality of $(5,27,32)$ is $\log(32)/\log(30)$. On One strong form of the ABC conjecture is that there are only finitely many triples (of relatively prime positive integers) with quality greater than $1.001$.

Now a solution to $x^m-y^n=k$ has $(A,B,C)=(k,y^n,x^m)$, and so $R\leq k y x$. Thus, the quality of the triple is at least $$\frac{m\log(x)}{\log(k)+\log(x)+\log(y)}\approx \frac{m\log(x)}{\log(k)+(\frac mn +1)\log(x)}.$$ As $x\to\infty$, this gives a quality approaching $\frac{mn}{m+n}\geq \frac{mn}{m+n}$, and for $m n>4$ this is $\geq 1.2$, which can only happen finitely many times by ABC. ActuallyAlso, $\frac{mn}{m+n}=1$ if $m=n=2$, so that case has to be handled separately.

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Pillai's conjecture is that for each $k$, there are only finitely many solutions.

The ABC conjecture implies Pillai's conjecture as follows. First, I state a form of the ABC conjecture. Given three relatively prime positive integers $A+B=C$, the quality of the triple $(A,B,C)$ is $\log(C)/\log(R)$, where $R$ is the product of the primes that divide $ABC$. For example, the quality of $(5,27,32)$ is $\log(32)/\log(30)$. On form of the ABC conjecture is that there are only finitely many triples (of relatively prime positive integers) with quality greater than $1.001$.

Now a solution to $x^m-y^n=k$ has $(A,B,C)=(k,y^n,x^m)$, and so $R\leq k y x$. Thus, the quality of the triple is at least $$\frac{m\log(x)}{\log(k)+\log(x)+\log(y)}\approx \frac{m\log(x)}{\log(k)+(\frac mn +1)\log(x)}.$$ As $x\to\infty$, this gives a quality approaching $\frac{mn}{m+n}\geq 1.2$, which can only happen finitely many times by ABC. Actually, $\frac{mn}{m+n}=1$ if $m=n=2$, so that case has to be handled separately.

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Pillai's conjecture is that for each $k$, there are only finitely many solutions.

The ABC conjecture implies Pillai's conjecture as follows. First, I state a form of the ABC conjecture. Given three relatively prime positive integers $A+B=C$, the quality of the triple $(A,B,C)$ is $\log(C)/\log(R)$, where $R$ is the product of the primes that divide $ABC$. For example, the quality of $(5,27,32)$ is $\log(32)/\log(30)$. On form of the ABC conjecture is that there are only finitely many triples (of relatively prime positive integers) with quality greater than $1.001$.

Now a solution to $x^m-y^n=k$ has $(A,B,C)=(k,y^n,x^m)$, and so $R\leq k y x$. Thus, the quality of the triple is $$\frac{m\log(x)}{\log(k)+\log(x)+\log(y)}\approx \frac{m\log(x)}{\log(k)+(\frac mn +1)\log(x)}.$$ As $x\to\infty$, this gives a quality approaching $\frac{mn}{m+n}\geq 1.2$, which can only happen finitely many times by ABC. Actually, $\frac{mn}{m+n}=1$ if $m=n=2$, so that case has to be handled separately.