make Mp prime are 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Comparison of this list with the correct data recorded in the top line of the table presented below shows that Mersenne made five mistakes. p = 67 and 257 do not yield prime values for Mp, and p = 61, 89 and 107 were not included in his list of special primes. With reference to explicit factoring, attention should be called to a val...

In the paper, making use of two inversion theorems for the Stirling numbers and binomial coefficients, the authors present an alternative explicit formula for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials. 1. M...