Zsigmondy's Theorem

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Zsigmondy's Theorem states that, for positive relatively prime integers $a$ , $b$ , and $n$ with $a>b$ , there exists a prime number $p$ (called a $\textit{primitive prime factor}$ ) such that $p$ $|(a^n-b^n)$ but $p\not|(a^k-b^k)$ for all positive integers $k<n$ EXCEPT (i) if $n=a-b=1$ , (ii) if $n=2$ and $a+b$ is a power of $2$ , or (iii) if $n=6$ , $a=2$ , and $b=1$ . This theorem can sometimes be used to prove that no more solutions exist to Diophantine equations.

Example

We desire to find all solutions $(x,y)$ to the Diophantine $3^x-1=5^y.$ We notice that the first integer $x$ for which $5|3^x-1$ is $x=4$ , which produces $3^4-1=80\neq 5^y$ . Now, by Zsigmondy's Theorem (the exceptions do not apply here), for any positive integers $n\geq 4$ , there must exist a primitive prime factor for each of the terms $3^n-1^n.$ However, the right hand side of the equation only contains prime factors of $5$ , and so there are $\boxed{\text{no solutions}}$ to this Diophantine equation.

Problems

Find all solutions $(x,y,z)$ to the Diophantine equation $7^x-1=2\cdot3^y\cdot5^z$ .

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Zsigmondy's Theorem

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