What is a Mersenne Prime Number?
The first 35 perfect numbers fit this same formula with “n” values of: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, and 1398269. The first eight are calculated and shown above. I leave the rest for you to calculate as an exercise in futility (just teasing). Another perfect number that has been found is equal to 2^2976220 * (2^2976221-1) If you multiplied this number out, you would have a number 1,791,864 digits long. We do not know if this would be the 36th, 37th, 38th or ??th perfect number. Since that discovery 2^3021376 * (2^3021377-1) has been found. Eventually, this record will be broken with an even larger number – and you could even be part of the team searching for that number).
A Mersenne prime number is a prime number which is one less than a power of two. About 44 have been discovered to date. For many years it was thought that all numbers of the form 2n – 1 were prime. In the 16th century, however, Hudalricus Regius demonstrated that 211 – 1 was 2047, with the factors 23 and 89. A number of other counter-examples were shown in the next few years. In the mid-17th century, a French monk, Marin Mersenne published a book, the Cogitata Physica-Mathematica. In that book, he stated that 2n – 1 was prime for an n value of 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257. At the time, it was apparent there was no way he could have tested the truth of any of the higher numbers. At the same time, his peers also could not prove or disprove his assertion. In fact, it wasn’t until a century later that Euler was able to demonstrate that the first unproven number on Mersenne’s list, 231 – 1, was in fact prime. A century later, in the mid-19th century, it was shown that 2127 –
