NextPrime[2^n] = NextPrime[3^m] has infinite groups of positive integer solutions I have a conjecture on next prime:
Conjecture (NextPrime Conjecture): Integer equation NextPrime[2^n] = NextPrime[3^m] on $m,n$ has infinite groups of positive integer solutions. When $2$ and $3$ are replaced by other two primes, it is also true!
In[59]:= t2 = Table[NextPrime[2^n], {n, 1, 200}];
t3 = Table[NextPrime[3^m], {m, 1, 190}];
t2 \[Intersection] t3


Out[59]= {5, 11}

My questions are:


*

*Do you think the NextPrime Conjecture is TRUE or FALSE?

*What is the third solution of NextPrime[2^n] = NextPrime[3^m], except 5==NextPrime[2^2]==NextPrime[3] and 11==NextPrime[2^3]==NextPrime[3^2]?
 A: According to the (unproven) Redmond-Sun conjecture, your conjecture is false. Indeed, for $n,m>1$ your condition is equivalent to there being no prime strictly between $2^n$ and $3^m$. Redmond-Sun conjecture states that a gap between two powers doesn't contain a prime only finitely many times.
If you believe that the list provided on Wikipedia is complete, then there isn't even the third number satisfying your condition.
A: We have $|2^x - 3^y| > 2^x e^{-\frac{x}{10}} > 2^x4^{-\frac{x}{10}} = 2^{\frac{4}{5}x}$ for x > 27
(See the link you posted in a comment. Unsolved Problems in Number Theory)
As an upper bound for prime gaps we have $g_n < p_n^\theta$ where $\theta = \frac{3}{4} + \epsilon$ for any $\epsilon > 0$ for n big enough. (See Wiki)
If n is big enough, we have for prime gaps near $2^x$: $g_n$ < $2^{x(\frac{3}{4} + \epsilon)}$. So if we take $\epsilon$ small enough and n big enough, the prime gap is smaller then the difference of powers of 2 and 3.
I didn't formulate it 100% rigorous. But I think the idea should be clear.
