Notation: $\ p_0=2,\ p_1=3,\ p_2=5, \ldots\ $ -- the increasing sequence of all primes.
(The following questions, once I've formulated them, remind me of Chebyshev). A very special case of a power $\pm 1\ $ would be $\ N^2-1 = p_m\cdot p_{m+1}\ $ -- a product of two prime twins. In general, let $\ m\in\mathbb Z_{\ge 0}\ $ and $\ d\ n\ N\in\mathbb N_{\ge 2}\ $ be such that $\ m<n.\ $ Finally, let
$$ N^d\pm 1\,\ =\,\ \prod_{k=m}^n p_k $$
Then primes $\ p_m\ \ldots\ p_n\ $ are called teammates, and $\ N^d\ $ is called a $\ (-1)$-container or $\ (+1)$-container depending on $\pm,\ $ or it can be called simply container. Also, $\ s:=n-m+1\ $ is called the team's size.
Example: Equation $$ 10^3+1 = 7\cdot 11\cdot 13 $$ presents a $(+1)$-container $\ 1000,\ $ for a size $\ 3\ $ team $\ 7\ 11\ 13.$
QUESTION: are there infinitely many teams of sizes $\ > 2\,?$
It'd be great to have any of the implicit questions answered too (I'll skip the routine task of asking them unless requested).
EDIT: let $\ m\in\mathbb Z_{\ge 0}\ $ and $\ d\ n\ N\,\ r\ M \in\mathbb N_{\ge 2}\ $ be such that $\ m<n.\ $ Finally, let
$$ N^d\pm 1\,\ =\,\ M^r\cdot\prod_{k=m}^n p_k $$
Then, following @GerryMyerson suggestive example, the sequence of consecutive primes $\ p_m\ldots p_n\ $ is called a GeMy near miss. Now we can ask about the infinitude of near misses (in addition to teams).
