Math: Double Checking

# DOUBLE CHECKING

To verify that a first-time Lucas-Lehmer primality test was performed without error, GIMPS runs the primality test a second time. During each test, the low order 64 bits of the final S(P)-2 value, called a residue, are printed. If these match, then GIMPS declares the exponent properly double-checked. If they do not match, then the primality test is run again until a match finally occurs.
GIMPS goes a bit further to guard against programming errors. Prior to starting the Lucas-Lehmer test, the S(0) value is left shifted by a random amount. Each squaring just doubles how much we have shifted the S value. Note that the mod 2p-1 step merely rotates the p-th bits and above to the least significant bits, so there is no loss of information. Why do we go to this trouble? If there were a bug in the FFT code, then the shifting of the S values insures that the FFTs in the first primality test are dealing with completely different data than the FFTs in the second primality test. It would be near impossible for a programming bug to produce the same final 64-bit residues.
Historically, the error rate for a Lucas-Lehmer test is a little over one percent.