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Conjecture: For all primes $p_n≥7$, there is at least one solution to the equation $$ p_n = p_k\# - p_m$$ where $p_k\#$ is any primorial, and $p_m$ is any prime number.

Has this been explored before? Can it been proven or disproven? I tried it numerically up to $ p_n =2322869$

Interestingly, the additive version of this, $$ p_n = p_k\# + p_m$$ seems to have solutions for most primes but not all; the lowest of which lacks a solution is $p_n=149$.

Daniel Gibson
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    why did you stop at 2322911 ? Seems there is no solution for this prime. – pietfermat Jul 31 '23 at 06:28
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    If that's true its an exciting result. I was using the first 51 primorials and didn't want to generate more and wait for it to factorize numbers well over three thousand digits. Did you check this using the first 171050 primorials? Because that sounds very computationally intensive – Daniel Gibson Jul 31 '23 at 06:36
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    It is easy to see why the additive version most likely don't have solution in many cases. Indeed, you only add positive numbers so you have a finite number of possibilities (and a very restricted number of posibilities because primorial grow very fast). But in the original equation, primorial can be as huge as you want, it can always be compensated by a larger prime since there is an infinity of prime numbers. – Cactus Jul 31 '23 at 07:08
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    No, I have not checked all 171050 primorials. But it's no coincidence you stopped at this prime. It would better to add this information to the question. – pietfermat Jul 31 '23 at 07:18
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    There are hard-cases (no solution upto $3000$#) under the mentioned limit. It would be nice to summarize them together with their solutions. Nevertheless +1 – Peter Jul 31 '23 at 08:12
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    The next hardcase (no solution upto $8\ 000$#) is $3\ 757\ 781$ – Peter Jul 31 '23 at 08:25
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    Following $4\ 128\ 689$ and $7\ 430\ 777$ – Peter Jul 31 '23 at 08:38
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    Following $14\ 023\ 523$ – Peter Jul 31 '23 at 08:44
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    factordb solves $3 757 781$ – Peter Jul 31 '23 at 08:55
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    factordb solves $4 128 689$ – Peter Jul 31 '23 at 09:00
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    Search limit for $2322911$ : $40\ 000$# – Peter Jul 31 '23 at 10:17
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    Search limit for $7 430 777$ : $25\ 000$# – Peter Jul 31 '23 at 10:18
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    factordb solves $14 023 523$ – Peter Jul 31 '23 at 10:28
  • Cactus, there is an upper limit to how big p_k# can be for a given p_n. There is a prime gap between p_k# - 1 and p_k# - p_(k+1), the latter being the upper bound for p_m. Therefore k+1 is less than or equal to n. Let me know if it is worth typing a full explanation for this – Daniel Gibson Jul 31 '23 at 11:21
  • Thanks, Peter. I didn't know about factordb – Daniel Gibson Jul 31 '23 at 11:22
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    I found a solution for $7430777$ , namely $48527$#$-7430777$ which is a probable prime. factordb has not yet confirmed it. – Peter Jul 31 '23 at 15:59
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    @Peter My bad. You are right. Earlier comment deleted. This is not whether every primorial can be decomposed, but whether every prime is featured in at least one decomposition. – Keith Backman Jul 31 '23 at 16:48
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    factordb – Peter Jul 31 '23 at 16:51
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    So, the only unsolved case upto $10^7$ is the original prime which has no solution upto $50\ 000$# – Peter Jul 31 '23 at 16:52
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    factordb – Peter Aug 01 '23 at 14:40

1 Answers1

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Partial answer : The above comments show that upto $14\cdot 10^6$ , there is a solution , the hardest case was the first prime that turned out to be tough. The primes are partially very big , so we will have to be content with probable primes.

What can we conclude now ? Hard to say. On the one hand , for every given prime $p>3$ , there are only finite many positive integers $n$ , for which $n$#$-p$ can be a prime number (we must have $n<p$ , otherwise $p$ is a nontrivial factor) and $n$# grows exponential.

On the other hand , the bigger the prime , the more chances we have to finally get a prime and that a prime factor of $n$#$-p$ with prime $p$ and $1<n<p$ must exceed $n$ increases the chance to get a prime , so it is hard to come to a final conclusion.

My guess is that there is a counterexample (a prime $p$ for which there is no solution) , but it might be very difficult to find it.

Peter
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