Find $p\in\mathbb{P}$ such that $p_{n,\lambda}=p+\lambda\cdot10630620+\displaystyle\sum_{r=1}^n2^r\in\mathbb{P}$ for $n=1,2,...,9$ and $\lambda=0,1$.

Question

Let $\mathbb{P}$ denote the set of prime numbers.

What is the smallest value of $p\in\mathbb{P}$ satisfying the following conditions:

1. For $n=1,2,3,...,9$, $$p_{n,0}=p+\sum_{r=1}^n2^r\in\mathbb{P}$$

2. For $n=1,2,3,...,9,$ $$p_{n,1}=p+10630620+\sum_{r=1}^n2^r\in\mathbb{P}$$

To reiterate the title:

Find the smallest value of $p\in\mathbb{P}$ such that $p_{n,\lambda}=p+\lambda\cdot10630620+\displaystyle\sum_{r=1}^n2^r\in\mathbb{P}$ for $n=1,2,...,9$ and $\lambda=0,1$.

This is a "puzzle" of my own making. I have the solution, but I'm curious to see other methods, especially ones that do not require brute force. Good luck!

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Kudos to dan_fulea for solving this!

2397347207, 2397347209, 2397347213, 2397347221, 2397347237, 2397347269, 2397347333, 2397347461, 2397347717, 2397348229
2407977827, 2407977829, 2407977833, 2407977841, 2407977857, 2407977889, 2407977953, 2407978081, 2407978337, 2407978849

Answer 1

In a third search...

sage: var('k');
sage: S = [ sum(2^k, k, 1, n) for n in [1..9] ]
sage: SJ = cartesian_product( [S, [0,1]] )
sage: for p in primes(10^9, 10^10):
....:     still_ok = True
....:     for s, j in SJ:
....:         q = p + s + j*10630620
....:         if not q.is_prime():
....:             still_ok = False
....:             break
....:     if still_ok:
....:         print p
....:         break
....:     
2397347207