More than 300 years ago Mersenne made pronouncements as to the prime or composite nature of 2P – 1 for all prime values of p from 1 to 257. His reasons were not disclosed, but the combined efforts of many mathematicians have shown that his statements were substantially correct. Of course when p is composite, two or more factors of 2p ± 1 will often be obvious but, as far as I am aware, only two general theorems relating to a non-obvious factor have been proved. I give these by way of introduction to this article and thereafter proceed to enunciate seven new and original theorems. As I offer no theoretical proofs of the theorems, they must be regarded as conjectures. They are, however, based on extensive computation which has engaged me for a period of three years.