Lucas-Lehmer test for Mersenne numbers (J)
According to the wikipedia article, first we define a sequence as follows:
Then a Mersenne number (Mp = 2p− 1) is prime iff
The sequence recurrence is straightfoward to define: NB. we define as a dyad so that we may do the modulo arithmetic on every iteration.
seq =: dyad : 'x|_2+*:y'
and, sure enough, the first few terms of the sequence agree with OEIS A00301. NB. 4x is 4 represented as an extended (arbitrary-precision) integer.
0 seq^:(i.8) 4x 4 14 194 37634 1416317954 2005956546822746114 4023861667741036022825635656102100994 16191462721115671781777559070120513664958590125499158514329308740975788034
Now we define a function, using the iteration operator (^:) to find the residue sp − 2(mod Mp):
res =: monad : '(_1+2x^y) seq^:(y-2) 4x'
 wrapping up
We can double check our implementation against the examples in the article:
res each 3 11 +-+----+ |0|1736| +-+----+ (0=res) each 3 11 +-+-+ |1|0| +-+-+
Or we can test primality in a different manner by counting (#) factors (q:) of the first few Mersenne numbers (Mp = 2p− 1), then checking that two sets of indications (zero residue or single factor) agree:
mpf =: monad : '#q:_1+2x^y' >((0=res)=(1=mpf)) each p:1+i.15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Alternatively, we could wrap everything up into the following single-line definition:
llmp =: monad : '0=(_1+2x^y) ([|_2+[:*:])^:(y-2) 4x'