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I'm looking for a set of prime numbers that satisfy the following property in the text below:
$p_{k+1}-p_k=p_k-p_{k-1}$,
$p_{k+i}-p_k=p_k-p_{k-i}$,
$p_{k-1},p_k,p_{k+1}$ consecutive prime numbers.

Example: $7-5=5-3=2$.

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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. $\endgroup$
    – Community Bot
    Aug 5, 2022 at 22:07

1 Answer 1

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The first requirement equates to asking for three consecutive prime numbers in an arithmetic progression. Three numbers in an arithmetic progression will have one member divisible by $3$ unless the difference between the members is divisible by $3$. Your example affords the only progression of primes that can have one member divisible by $3$.

Since primes greater than $3$ have the form $6m\pm 1$, the difference between the consecutive primes will have to be a multiple of $6$. These are not rare. For example, $47,53,59$.

Using that as a starting point, where $p_{16}=53$ we can find $p_{9}=23$ and $p_{23}=83$, representing $i=7$ in the second condition, with a common difference of $30$.

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