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Let $E$ be a positive even integer. Define the following:

Let
$$ P_E := \{ p \in \text{Primes} \mid 3 \leq p < E/2 \} $$
i.e., the set of odd primes less than $E/2$.

Define the odd primorial of $E/2$ as:
$$ \mathrm{Primorial}_{\mathrm{odd}}(E/2) := \prod_{p \in P_E} p $$

Define:
$$ T := E + \mathrm{Primorial}_{\mathrm{odd}}(E/2) $$

Question:

Does there exist a positive even integer $E$ such that every integer in the interval
$$ [T - E + 1,\ T] $$
is composite?

That is, the interval of width $E$, ending at $T$, consist entirely of composite numbers?

Provide if possible a rigorous proof or disproof of the existence of such an $E$ . Any insights from sieve methods, properties of prime gaps, or known bounds on consecutive composite numbers would be especially appreciated.

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    $\begingroup$ Well, the natural first step is to conduct a search. How far out have you looked? What sort of behavior have you observed? $\endgroup$ Commented May 21 at 23:01
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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$ Commented May 21 at 23:22

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$E=88$.
Then $\frac{E}{2}=44$, so $$\mathrm{Primorial}_{\mathrm{odd}}(44)=3 \cdot 5 \cdot 7 \cdot 11 \cdots 41 \cdot 43 = 6,541,380,665,835,015$$ So $$T=6,541,380,665,835,103 \Rightarrow I=[6,541,380,665,835,016,\ 6,541,380,665,835,103]$$ and I've checked manually, none of the numbers in this interval are prime. Other than that my code didn't find anything up to $1000$. I have no idea how to prove $88$ is the only number.

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  • $\begingroup$ Thanks for running a program. Can I just verify what range you checked for primes in? The range should be [T-E+ 1 ,T] I was expecting a possible E to be much larger than this. $\endgroup$ Commented May 21 at 23:56
  • $\begingroup$ @FelixFowler It's only up to 1000 smallish numbers, so it wasn't very hard to just plug the interval itself into a prime number sieve instead of checking for prime numbers in that range. $\endgroup$ Commented May 21 at 23:59

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