# 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:

1. Do you think the NextPrime Conjecture is TRUE or FALSE?

2. 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]?

• Do you have any specific reason to believe that this might be true? Do you have any ideas about how you might prove it? What makes this more than an idea restated as a conjecture? Nov 28, 2015 at 8:36
• @Mark Bennet no specific reason, no idea to prove . I think this question is as hard as Riemann hypothesis Nov 28, 2015 at 9:07

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.

• Yes，NextPrime Conjecture is fighting with Redmond-Sun conjecture, But, at whose hand will the deer die? I can't accept a reference for answer. In my opinion, Redmond-Sun conjecture is FALSE, which only verified below 10^12. Nov 28, 2015 at 9:28
• @aboy From what you have said in question body, you have even less evidence for your own conjecture. Checking numbers up to exponent $200$ and finding only two examples isn't a good evidence for infinitude. Nov 28, 2015 at 12:46
• @aboy Right now I'm asking out of pure curiosity, but do you have any reason to insist that NextPrime conjecture should be true? Nov 28, 2015 at 13:12
• It's relatively easy to disprove the "NextPrime Conjecture", using standard lower bounds for $|3^x-2^y|$ from linear forms in logarithms (as in de Weger's thesis) and upper bounds for gaps between primes. Nov 28, 2015 at 17:53
• @aboy - insisting is irrelevant. Proof or at least convincing numerical evidence. Nov 29, 2015 at 16:54

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.