A reference implementation of a recent variant can be found in this prior Community thread.

It is certainly possible to implement what you want, but it takes a lot of time and extensive programming knowledge.I not a programer with this skills.

Code is slow and naive. Using: Aria AI in Opera borwser give me:

 LucasLehmerTest[n_Integer] := 
  Module[{s = 4, i}, 
   For[i = 3, i <= n, i++, s = Mod[s^2 - 2, 2^n - 1];];
   If[s == 0, True, False]]

LucasLehmerTest[127]
(*True*)

primes = {3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701};
Table[LucasLehmerTest[primes[[j]]], {j, 1, Length@primes}] // AbsoluteTiming

(*{6.71683, {True, True, True, True, True, True, True, True, True, True,
   True, True, True, True, True, True, True, True, True, True, True, 
  True, True, True}}*)

Regards M.I.