The largest prime was discovered eighteen days ago by a mathematician named Curtis Cooper. The specific prime that he found was a Mersenne prime which contains exactly digits. Although found by chance, there was some mathematics involved that computers used to determine if a certain number was prime or not.
It is called the Lucas-Lehmer “Theorem” (better off as a “sequence”) in which we start from the number with the next number being . So the sequence would go and it will continue to rapidly increase exponentially.
Let us consider the example which is indeed prime. To verify, we take the exponent, , and subtract to get (). We then take a look at the second term in the Lucas-Lehmer “sequence” which is . We need to show that is divided with remainder by the “prime” number . So
And indeed we receive a remainder of therefore confirming that is prime.
So let such that and the Lucas-Lehmer “sequence” be denoted by . Then for a given “testable” number, it is prime iff
Otherwise, not prime.
Today’s post is short because I have finals this week. More to come next week!