Lucas-Lehmer Test:

for 



p


{\displaystyle p}

 an odd prime, the Mersenne number 




2

p


−
1


{\displaystyle 2^{p}-1}

 is prime if and only if 




2

p


−
1


{\displaystyle 2^{p}-1}

 divides 



S
(
p
−
1
)


{\displaystyle S(p-1)}

 where 



S
(
n
+
1
)
=
(
S
(
n
)

)

2


−
2


{\displaystyle S(n+1)=(S(n))^{2}-2}

, and 



S
(
1
)
=
4


{\displaystyle S(1)=4}

.



Task

Calculate all Mersenne primes up to the implementation's
maximum precision, or the 47th Mersenne prime   (whichever comes first).




Select[Table[M = 2^p - 1; 
  For[i = 1; s = 4, i <= p - 2, i++, s = Mod[s^2 - 2, M]]; 
  If[s == 0, "M" <> ToString@p, p], {p, 
   Prime /@ Range[300]}], StringQ]
 
=> {M3, M5, M7, M13, M17, M19, M31, M61, M89, M107, M127, M521, M607, M1279}