Fortunate Primes

Conjecture (Original)

A conjecture made by Reo F. Fortune that if $E_k$ = $p_k$# + 1, where $p_k$# is the $k^{th}$ prime and p# is primorial, and assuming $q_k$ is the next prime after $E_k$ (i.e., the smallest prime greater than $E_k$), then $F_k$ = $q_k$ - $E_k$ + 1 is prime for all k. The first values of $F_k$ are 3, 5, 7, 13, 23, 17, 19, 23, ….

Conjecture

Let S₀ = $p_k$# + $p_{k+1}$. If $S_0$ is composite, let $q_0$ be the smallest prime dividing $S_0$. Let $S_i = p_k# + q_{i-1}$, where $q_{i-1}$ is the smallest prime dividing $S_i$.

Assuming $S_0 \lt M^2$, then $S_i \lt {M + 1}^2 \forall$ i. Then $S_n$ is prime for some n ≥ 0. Otherwise, $S_i$ = $S_j$ for some 0 ≤ i < j.