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Suppose $a$ is a proper divisor of $n$ (where $n$ is a positive integer), and $b$ a proper divisor of $n + 1$.

Is there a general criterion (or general property of $n$) which enables one to conclude that $a - b$ cannot be divisible by at least two distinct prime divisors of $n$ (whatever $a$ and $b$ are) ?

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  • $\begingroup$ What about $a=7$, $b=13$, $n=168$? $\endgroup$
    – YCor
    Commented Dec 6 at 9:20
  • $\begingroup$ @YCor: what about it ? $\endgroup$
    – THC
    Commented Dec 6 at 9:34
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    $\begingroup$ I think the question is far too vague. For instance, $n$ being a prime power is a trivial sufficient criterion ... $\endgroup$
    – Peter Mueller
    Commented Dec 6 at 11:43
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    $\begingroup$ I see. Sounds a bit like "what is a general criterion to conclude that a number $n$ is even?" $\endgroup$
    – YCor
    Commented Dec 6 at 13:00

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