The five primes, 131, 157, 211, 349, 739, are neither in arithmetic or geometric progression, but are instead the sum of the five corresponding terms of an arithmetic and geometric progression. 


Are there arbitrarily long sequences of primes with this property, that is, that they are the sum of the corresponding terms of an arithmetic and a (non-constant) geometric progression?


<https://puzzling.stackexchange.com/questions/89739/primes-from-arithmetic-and-geometric-progressions/89740#89740>



Primes from arithmetic and geometric progressions
Claim: I actually have almost zero knowledge in this domain, and this answer is totally based on the correctness of [this wiki page](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem). Moreover, it could be that I misunderstood the content of that page, so it would be kind of some expert to point out to me if I made a mistake.


Quote from that wiki page:



> 
> More precisely, given any integer-valued polynomials $P\_1,\cdots, P\_k$ in one unknown $m$ all with constant term $0$, there are infinitely many integers $x$, $m$ such that $x + P\_1(m), \cdots, x + P\_k(m)$ are simultaneously prime.
> 
> 
> 


From what I understand, this implies (or means) that there exists **positive** integers $x$ and $m$ such that all the $x + P\_i(m)$ are primes.


Now for any fixed $k$, simply take $P\_i(m) = m^i + im$. We will then have:
$$x + P\_i(m) = (x + im) + m^i,$$
where $(x + im)\_i$ and $(m^i)\_i$ are arithmetic and geometric progressions, respectively.


