Can we prove the following statement ?

Every natural number $n$ can be written as $n=s-t$ ($s,t$ positive integers) with $\omega(s)=\omega(t)$ , in other words , the difference of two positive integers with the same number of distinct prime factors.

  • If $n=1$ , we can choose $\ s=3\ $ and $\ t=2$
  • If $n$ is even , we can choose $\ s=2n\ $ and $\ t=n$

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    $\begingroup$ What is the question? $\endgroup$ – mathreadler Jan 11 at 8:30
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    $\begingroup$ If $n$ is even, then $2n$ and $n$ have the same number of prime factors, so $n = 2n-n$ is a solution $\endgroup$ – Charles Madeline Jan 11 at 8:39
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    $\begingroup$ @CharlesMadeline At least we can be sure that a counterexample , if there is actually one, must be huge. $\endgroup$ – Peter Jan 11 at 9:19
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    $\begingroup$ We have other possibilities. If we find a number $m$, such that $m$ and $m+1$ are both coprime to $n$ and have the same number of prime factors, we have found a solution as well. $\endgroup$ – Peter Jan 11 at 9:21
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    $\begingroup$ @CharlesMadeline Moreover, we only concentrated on solutions of the form $(k+1)n-kn$. This is not required in the question. But I agree that the proof is not yet finished. $\endgroup$ – Peter Jan 11 at 9:52

There is a way to prove this directly though, and it is pretty simple:

Case 1: Both 2 and 3 divide $n$, or neither 2 nor 3 divide $n$: Then let $s=3n$ and $t=2n$.

Case 2: 3 does not divide $n$ but $2$ does: Then let $s=2n$ and $t=n$.

Case 3: 3 divides $n$ but 2 does not: Then let $p$ be the smallest odd prime that does not divide $n$. Then let $s=pn$ and let $t=(p-1)n$. [Note that $p-1$ is the product of smaller odd primes, all of which divide $n$ [by def'n of $p$], and 2, which does not. So $\omega(pn) = \omega(n) +1$ [because $p$ doesn't divide $n$], while $\omega((p-1)n) = \omega(n) +1$ as well, because the one prime factor of $p-1$ that does not divide $n$ is 2.]

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    $\begingroup$ Nice solution ! $\endgroup$ – Kolja Jan 12 at 0:22

The Bunyakovsky-conjecture implies that we always find a solution.

To show this, assume that there are infinite many positive integers $p$, such that $$2p+1$$ $$2p+3$$ $$2p^2+4p+1$$ are simultaneously prime. Then, we can choose $p>n$ satisfying this property. Then, $$(2p+1)(2p+3)n-2(2p^2+4p+1)n=n$$ is a solution, whenever $n$ is odd (the even case has already been solved).

There are plenty of other possibilities to choose the expressions, so the given statement is in fact much weaker than Bunyakovsky's conjecture.


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