# Modularity and prime number sequence

I tried to solve this modular equation involving first $n$ prime numbers. And it is:

$$2^{3+5+7+11+13+…..+p_{n-3}+p_{n-2}}\equiv p_{n-1}\ \left(\text{mod }p_{n}\right),$$

where $p_{n}$ is the $n$-th prime number.

I couldn’t find any solution for this equation until first $300$ primes. Is there any solutions for $n$?

#### Solutions Collecting From Web of "Modularity and prime number sequence"

I found solutions at $p_{n-1}=5813, 29537,$ and $44839$ [$p_n = 5821, 29567,$ and $44843$ respectively]. We’re hitting a fairly random point in the power cycle so I see every reason to expect solutions to continue to appear occasionally.

For what is worth, next solutions $\{n,p_n\}$ are:
$$\{306\,311,4\,353\,467\}\\ \{859\,825,13\,174\,621\}\\ \{1\,700\,098,27\,291\,793\}\\$$
There are no more with $n\le10^7$.