# I'm looking for a set of prime numbers that satisfy the following property in the text below [closed]

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

• 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.
– Community Bot
Aug 5, 2022 at 22:07

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 $$30$$.