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Prove that any integer can written as differences of two numbers with same prime divisors.

The problem seems wrong since we can prove that it is impossible for $1$ and any prime bigger than $2$.I need the original problem but not an answer since I want to think on it.Maybe we should prove it for any composite number.

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  • $\begingroup$ @DietrichBurde I am sure about the part two numbers with same prime divisors. $\endgroup$ Commented Oct 20, 2017 at 15:44
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    $\begingroup$ Odd numbers cannot be written as the difference of two numbers with same prime divisors since $\text{odd}=\text{even}-\text{odd}$ or $\text{odd}=\text{odd}-\text{even}$. Even numbers can be written as the difference of two numbers with same prime divisors since $2m=4m-2m$. $\endgroup$ Commented Oct 20, 2017 at 17:07

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This statement cannot hold for all composite numbers. A simple example is $n=9$. Actually, $n$ is a difference of two numbers with same prime divisors if and only if $n$ is even - see mathlove's argument above.

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