Four Fortune-Type Prime Conjectures: Primorials, Least Common Multiples, and Logarithmic Windows

This note presents four conjectures concerning the primality of minimal offsets in prime-generating expressions, extending the framework of the classical Fortunate number conjecture. The first is the classical Fortunate conjecture, which defines \(F_n\) as the smallest integer \(m\ge 2\) such that \(p_n\#+m\) is prime, and asserts that \(F_n\) is always prime. The second introduces an LCM-based construction, defining \(G_n\) as the smallest \(m\ge 2\) for which \(\operatorname{lcm}(1,\ldots,n)+(\operatorname{lcm}(1,\ldots,n)+1)m\) is prime. The third considers a next-prime-weighted primorial construction, defining \(H_n\) as the smallest \(m\ge 2\) such that \(p_n\#+p_{n+1}m\) is prime. The fourth proposes a logarithmic-window construction using \(k_n=\lceil\log_2(n+1)\rceil^2\) and \(A_n=\prod_{j=0}^{k_n-1}(n+j)\). It defines \(J_n\) as the smallest \(m\ge 2\) for which \(A_n+(A_n+1)m\) is prime. Unlike full factorial or primorial constructions, this anchor requires only \(O((\log n)^2)\) factors. A common divisibility argument establishes that successful composite offsets must exceed explicit quadratic thresholds determined by excluded small prime factors. The note derives sufficient conditions for primality, provides illustrative computations, and discusses computational complexity and possible avenues toward proof. All four prime-offset assertions remain unproved. The classical Fortunate conjecture is established prior work, while the originality of the three proposed variants has not been conclusively verified.