You can find some amusing papers in this area by searching for ["primes at a glance"][1] and [" primes at a (somewhat lengthy) glance"][2]. $31=2^23^2-5$ 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 $31$.)


  [1]: http://www.jstor.org/stable/2007883%20
  [2]: http://www.dms.umontreal.ca/~andrew/PDF/agoh.pdf