1
$\begingroup$

I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:

Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$ contain a product of exactly two distinct primes.

First, let $|\cdot|_2$ denote the bitsize of a number.

This result lead me as follows: For a fixed $m$ and any semiprime $n=pq$ such that $|n|_2=m$, what is the probability that $|p|_2=|q|_2$?

Or, similarly, for a given $x$ how far would we have to walk up the number line to guarantee that we will find such an $n$? That is: for what function $f(x)$ can we guarantee that the interval $[x, x + f(x)]$ contains such an $n$?

Edit 1: The bitsize of a number is the number of digits in the binary representation of a number. In particular, this is how computers represent numbers in hardware. For example, the bitsize of $33$ is $6$ because $33 = (100001)_2$ in base $2$.

Improve this question
$\endgroup$
4
  • $\begingroup$ What is the bitsize of a number? $\endgroup$
    – Joshua Stucky
    Feb 1, 2022 at 0:28
  • 1
    $\begingroup$ @JoshuaStucky I added this clarification as an edit. $\endgroup$
    – factorn
    Feb 1, 2022 at 2:30
  • $\begingroup$ The formula $\lceil \log_2 x\rceil$ is perhaps an easier way to describe the bitsize of a number, although generally people don't bother with the $\lceil \cdot \rceil$. $\endgroup$
    – Mark Schultz-Wu
    Feb 1, 2022 at 6:49
  • 2
    $\begingroup$ That being said, the answer appears to be not that common, see here $\endgroup$
    – Mark Schultz-Wu
    Feb 1, 2022 at 6:54

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.