3 problems

Let $\mathcal{D}_k^1=L_{\infty}(p_{k+1})-L_{\infty}(p_k)$ be the prime gaps along the number trail, and let $\mathbb{N}$ denote the natural numbers. Modified Polignac's conjecture.…

Let $\pmb{\omega}=(\omega_0,\omega_1,\ldots,\omega_k)\notin\mathcal{F}$ be a finite-length word that contains no $\pmb{\tau}\in\mathcal{F}$ as a subsequence of consecutive symbols.…

Let $\mathbf p=(p_1,\ldots,p_n)$ be the ordered tuple of primes used to define the sequence $\pmb{\Omega}$, and let $\pmb{\omega}$ be a finite word that occurs in $\pmb{\Omega}$. P…