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

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

QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the equation: $$|O_p-2^d|=1$$ in natural numbers $d$, and odd primes $p$?

(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 $\ p$.

ACKNOWLEDGEMENT: Bjørn Kjos-Hanssen has provided one of the above solutions (see his answer below).

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

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

QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the equation: $$|O_p-2^d|=1$$ in natural numbers $d$, and odd primes $p$?

(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 $\ p$.

ACKNOWLEDGEMENT: Bjørn Kjos-Hanssen has provided one of the above solutions (see his comment below).

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

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

QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the equation: $$|O_p-2^d|=1$$ in natural numbers $d$, and odd primes $p$?

(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 $\ p$.

ACKNOWLEDGEMENT: Bjørn Kjos-Hanssen has provided one of the solutions (see his answer below).

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

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

QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the equation: $$|O_p-2^d|=1$$ in natural numbers $d$, and odd primes $p$?

(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 $\ p$.

ACKNOWLEDGEMENT: Bjørn Kjos-Hanssen has provided one of the above solutions (see his answer below).

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

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

QUESTION: Are the ordered pairs $(d\ p)=(1\ 3)$ and $(d\ p)=(4\ 5)$ the only solutions of the equation: $$|O_p-2^d|=1$$ in natural numbers $d$, and odd primes $p$?

(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 $\ p$.

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

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

QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the equation: $$|O_p-2^d|=1$$ in natural numbers $d$, and odd primes $p$?

(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 $\ p$.

ACKNOWLEDGEMENT: Bjørn Kjos-Hanssen has provided one of the solutions (see his answer below).