Primality proof for n = 1716199415032652428745475199770348304317358825035826352348615864796385795849413675475876651663657849636693659065234142604319282948702542317993421293670108523:

Take b = 2.

b^(n-1) mod n = 1.

22678127291974494752999082758512529677847838471000074610295040629304883810213774415831891245381957284486015221359371338545320928049961 is prime.
b^((n-1)/22678127291974494752999082758512529677847838471000074610295040629304883810213774415831891245381957284486015221359371338545320928049961)-1 mod n = 1195096767381846247866902946820763906702983690002105628954172954786737946583034188187632453921668214043256957401586659298281779806004081275451771513884139081, which is a unit, inverse 1376988227778942974628325006910981359520723527903109584916037024961358907392043319728802611191048154009726299542091615960946726317982618295265480520016070706.

(22678127291974494752999082758512529677847838471000074610295040629304883810213774415831891245381957284486015221359371338545320928049961) divides n-1.

(22678127291974494752999082758512529677847838471000074610295040629304883810213774415831891245381957284486015221359371338545320928049961)^2 > n.

n is prime by Pocklington's theorem.