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).




module Main
  where
 
main = printMersennes $ take 45 $ filter lucasLehmer $ sieve [2..]
 
s mp 1 = 4 `mod` mp
s mp n = ((s mp $ n-1)^2-2) `mod` mp
 
lucasLehmer 2 = True
lucasLehmer p = s (2^p-1) (p-1) == 0
 
printMersennes = mapM_ (\x -> putStrLn $ "M" ++ show x)