New explorations and remarkable inequalities related to Fortune’s conjecture and fortunate numbers

Fortune's conjecture (named after the social anthropologist Reo Franklin Fortune) is an extremely elegant mathematical conjecture that always remains an open problem in number theory. It is a conjecture about prime numbers, which leads to the so-called "fortunate numbers" (not to be confused with "lucky numbers"): Reo F. Fortune predicted that no fortunate number is composite. This conjecture impresses us all as mathematicians, that's why we decided that it will be the subject of this paper, which has many objectives and very interesting findings, among them: - Highlighting numerous properties of the fortunate numbers. - Giving a proof of Fortune's conjecture in a particular case, by two different methods one of which is original. - Presenting many counterexamples that reinforce the previous point, when the satisfied hypothesis in that particular case, is not met. - Proving a new remarkable inequality that all the 3000 first (known until now) fortunate numbers perfectly fulfill. Despite our continuous research (quite recently) on the subject of Fortune’s conjecture, we have never found a mathematical reference with a variety of ideas dealing with this conjecture, so we hope that this paper will be the first scientific work containing multiple ideas, comments, results and goals, and contributing significantly to find a definitive solution of Fortune's conjecture, therefore this paper may advance the field.