See here https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test for the definition of a strong probable prime, where it is also mentioned that Arnault constructed a number being strong probable prime to many bases (A version of maple declared the number to be prime!)
Is there an easy way to construct such numbers, for example a number strong probable prime to the bases $2-1000$ ? It would already be nice to see Arnaults number (I did not find it online), but I would like to construct my own numbers.
A promising approach are the numbers of the form $N=(4k-1)(8k-3)$, where both $4k-1$ and $8k-3$ are prime. It can be shown that such numbers have the minimum number of witnesses (exactly $75$% of the numbers coprime to $N$), but it is difficult to make the smallest witness large.
Can anyone help ?
Additional question : If $p$ is a prime, is there a known estimate for the smallest number which is strong probable prime to every base $q$ with $q$ prime and $q\le p$ (I do not demand the number to be strong-probable prime to every base $\le p$) ?