7
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I have tried a number of methods using prime factorisations but they inevitably lead to invoking too many unknowns for me and balloon in complexity.

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  • $\begingroup$ do u have any example for such numbers? $\endgroup$
    – Kiran
    Commented Jan 26, 2017 at 17:05
  • $\begingroup$ I do have some, and they were quite a challenge to find. $\endgroup$
    – Ivan Neretin
    Commented Jan 26, 2017 at 18:34
  • 1
    $\begingroup$ can u post any such numbers, just for curiosity, $\endgroup$
    – Kiran
    Commented Jan 26, 2017 at 18:52
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    $\begingroup$ This is an old problem of Erd\H{o}s, referenced in Guy's Unsolved Problems in Number Theory (D13). Chao Ko found infinitely many solutions, the smallest of which has $x=12^6$, $y = 6^8$ and $z=2^{11} 3^7$. $\endgroup$
    – Mike Bennett
    Commented Jan 28, 2017 at 3:22

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