You can find some amusing papers in this area by searching for "primes at a glance" and " primes at a (somewhat lengthy) glance". $31=2^23^2-5$ is enough to show $31$ has no prime divisors below its square root.
$A=88711$ is the product of $7,19,23,29$ and $72930$ is the product of $2,3,5,11,13,17$ so we can certainly find positive coprime integers $x,y$ with $1=88711x-72930y.$ Then $31=88711s-72930t$ is a difference of coprime values for $s=31x+72930$ and $t=31y+88711$ You can always do that. But probably not with $st$ having all prime factors below 31 ( in which case the prime factors would split the same way, given that nome of them divide $31$.)