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Let $p$ be large prime in $[T,2T]$ where $T$ is a parameter.

  1. Can we have an integer pair $a,b$ satisfying $ab\equiv c\bmod p$ such that $|c|$ is of size $O(\operatorname{polylog}(T))$ and $a,b$ are of size $O(T^{1/2 +\varepsilon})$?

  2. If $1/2$ is impossible what is the best rational in exponent we can get?

  3. Given $p$ how many such minimal generically positioned pairs are possible?

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  • $\begingroup$ A good person to ask would be Igor Shparlinski $\endgroup$
    – Reherueywyryökrdjjd
    Commented Apr 23, 2022 at 20:06
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    $\begingroup$ It is believed that there are infinitely many primes of the form $n^2-2$. For such a prime, let $a=b=n$, $c'=2$. (Why $c'$, and not just $c$?) (Is $c'$ allowed to be negative?) $\endgroup$
    – Gerry Myerson
    Commented Apr 24, 2022 at 1:58
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    $\begingroup$ In particular, the following survey of Shparlinski has a lot of information about this sort of problem: arxiv.org/abs/1103.2879 $\endgroup$
    – Thomas Bloom
    Commented Apr 24, 2022 at 9:28

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There is a nice article of Heath-Brown in the Mathematical Intelligencer, called Arithmetic applications of Kloosterman sums, where this problem is discussed under the heading, "An elementary problem." He derives exponent $3/4$ and remarks that it is open to improve on it. I am not aware of any subsequent improvements.

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