**NOTATION**: $O_x$ -- the product of all odd primes $\le x$.

E.g.  $O_7=3\cdot 5\cdot 7 = 105$.

**QUESTION**:  Are the order pairs  $(d\ p)=(1\ 3)$ and $(d\ p)=(4\ 5)$ the only solutions of the equation:
$$|O_p-2^d|=1$$
in a natural number $d$, and an odd prime $p$?

(I don't know an answer).

**MOTIVATION**:  Let $\ s\ $ be a prime just after $\ p$, so that $\ p<s$. If
$$|O_p-2^d|\ne 1$$
then
$$|O_p-2^d|\ge s$$
Furthermore, sometimes $\ s\ $ can be quite a bit larger than $\ p$.