Math: Lucas-Lehmer Details |
The Lucas-Lehmer primality test is remarkably simple. It states that for P > 2, 2P-1 is prime if and only if Sp-2 is zero in this sequence: S0 = 4, SN = (SN-12 - 2) mod (2P-1). For example, to prove 27 - 1 is prime:
S0 = 4
S1 = (4 * 4 - 2) mod 127 = 14
S2 = (14 * 14 - 2) mod 127 = 67
S3 = (67 * 67 - 2) mod 127 = 42
S4 = (42 * 42 - 2) mod 127 = 111
S5 = (111 * 111 - 2) mod 127 = 0
To implement the Lucas-Lehmer test efficiently, one must find the fastest way to
square huge numbers modulo 2P-1. Since the late 1960's the fastest algorithm for
squaring large numbers is to split the large number into pieces forming a large
array, then perform a
Fast Fourier Transform (FFT), a squaring, and an Inverse
Fast Fourier Transform (IFFT). See the "How Fast Can We Multiply?" section in
Knuth's Art of Computer Programming vol. 2. In a January, 1994 Mathematics of
Computation article by Richard Crandall and
Barry Fagin titled "Discrete
Weighted Transforms and Large-Integer Arithmetic", the concept of using an
irrational base FFT was introduced. This improvement more than doubled the speed
of the squaring by allowing us to use a smaller FFT and it performs the mod 2P-1
step for free. Although GIMPS uses a floating point FFT for reasons specific to
the Intel Pentium architecture, Peter Montgomery showed that an all-integer
weighted transform can also be used.
As mentioned in the last paragraph, GIMPS uses floating point FFTs written in
highly pipelined, cache friendly assembly language. Since floating point
computations are inexact, after every iteration the floating point values are
rounded back to integers. The discrepancy between the proper integer result and
the actual floating point result is called the convolution error. If the
convolution error ever exceeds 0.5 then the rounding step will produce incorrect
results - meaning a larger FFT should have been used. One of GIMPS' error checks
is to make sure the maximum convolution error does not exceed 0.4.
Unfortunately, this error check is fairly expensive and is not done on every
squaring. There is another error check that is fairly cheap. One property of FFT
squaring is that:
(sum of the input FFT values)2 = (sum of the output IFFT values)
Since we are using floating point numbers we must change the "equals sign" above
to "approximately equals". If the two values differ by a substantial amount,
then you get a SUMINP != SUMOUT error as described in the readme.txt file. If
the sum of the input FFT values is an illegal floating point value such as
infinity, then you get an ILLEGAL SUMOUT error. Unfortunately, these error
checks cannot catch all errors, which brings us to our next section.
What are the chances that the Lucas-Lehmer test will find a new Mersenne prime
number? A simple approach is to repeatedly apply the observation that the chance
of finding a factor between 2X and 2X+1 is about 1/x. For example, you are
testing 210000139-1 for which trial factoring has proved there are no factors
less than 264. The chance that it is prime is the chance of no 65-bit factor *
chance of no 66 bit factor * ... * chance of no 5000070 bit factor. That is:
64 * 65 * .... * 5000069
65 66 5000070