For the musement of the trade alone, and not weary about real-life, in the spirit of GH Hardy, who invites us to do math as a fine art, I'm interested in constructing "random walks" on the prime numbers with certain "ergodic" properties. I've reduced my troubles to the following simple --to understand, not to solve!-- conjecture. Viz,
Let $p_1 < p_2 < p_3 < \ldots < p_n < \ldots$ be the sequence of primes (with $p_1 := 2$ as usual).
Conjecture: There are infinitely many positive integers $n$ for which
\begin{eqnarray}
p_n + p_{n + 3} - 2 p_{n + 2} = 0.
\end{eqnarray}
For example, the above relation holds for $n = 2$ since $p_2+ p_5 - 2p_4 = 3 + 7 - 2 \times 5 = 0$. Follow technical thread here.
