What is this shortcut to determine primality? I'm watching this, he says that David Slowinski discovered the biggest prime in 1984: $2^{132,049}$-1 and that it took 1 week on a Cray supercomputer: using some shortcut and that the absence of this shortcut would make the process last the age of the universe. What's this shortcut?

Numbers of the form $2^n-1$ can be tested by the Lucas-Lehmer method, which is not available for numbers not of that form.