In this post we denote the sequence of prime numbers as $p\_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.


**Definition.** *We define the* $\theta$*-strong primes, or strong primes at level* $\theta$, *as the sequence of those prime numbers* $p\_n$ *that satisfy the inequality* $$p\_n>\theta\, p\_{n-1}+(1-\theta)p\_{n+1}\tag{1}$$
*for some fixed real number* $0<\theta<1$.


**Remark.** My definition arises as a generalization of the known as strong primes. See the definition in number theory of strong primes from the Wikipedia [*Strong prime.*](https://en.wikipedia.org/wiki/Strong_prime#Definition_in_number_theory)



> 
> **Question.** I would like to know, if my question is interesting, if it is possible to prove that there exists some $\hat\theta$ for which the corresponding sequence of prime numbers has finitely/infinitely many terms. **Many thanks.**
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Here I emphasize that $0<\theta<1$.


I have no intuition if my question is interesting, I don't know what can be a more interesting question at research level about the study of these sequences $(1)$ and their corresponding $\theta$. I did some calculations for $\theta\neq\frac{1}{2}$ a rational number.



A generalization of strong primes
Regarding any $\hat\theta$ for which the prime numbers sequence has finitely/infinitely many terms, consider one which has only finitely many terms. There would then exist a prime index $m$ for which all $n \gt m$ gives


$$p\_n \le \hat\theta\, p\_{n-1} + (1 - \hat\theta)p\_{n+1} \tag{1}$$


Using the standard definition of prime gaps of


$$g\_n = p\_{n+1} - p\_{n} \tag{2}\label{eq2A}$$


we then have


$$\begin{equation}\begin{aligned}
p\_n & \le \hat\theta(p\_{n} - g\_{n-1}) + (1 - \hat\theta)(p\_{n} + g\_{n}) \\
p\_n & \le \hat\theta p\_{n} - \hat\theta g\_{n-1} + (1 - \hat\theta)p\_{n} + (1 - \hat\theta)g\_{n} \\
\hat\theta g\_{n-1} & \le (1 - \hat\theta)g\_{n} \\
g\_n & \ge \left(\frac{\hat\theta}{1 - \hat\theta}\right)g\_{n-1}
\end{aligned}\end{equation}\tag{3}\label{eq3A}$$


If $\hat\theta \gt 0.5$, then $\frac{\hat\theta}{1 - \hat\theta} \gt 1$, so \eqref{eq3A} shows the prime gaps are strictly increasing for all $n \gt m$. However, this contradicts that there are infinitely many prime gaps of at most $246$ (e.g., see [Bounded gaps between primes](https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes)). If $\hat\theta = 0.5$ instead, then $\frac{\hat\theta}{1 - \hat\theta} = 1$, so the prime gaps are non-decreasing, but this is also not possible since prime gaps can become arbitrarily large and, thus, must decrease later to become the smaller prime gaps, e.g., those at most $246$.


This means the original assumption of there being only a finite number of primes in the sequence must be incorrect, i.e., there are infinitely many $\theta$-strong primes for *any* $\hat\theta \ge 0.5$.


**Update:** The PDF version of the arXiv article [On the ratio of consecutive gaps between primes](https://arxiv.org/abs/1406.2658), near the bottom of page $8$, states about Erdős



> 
> He mentioned 60 years ago [Erd5]: "One would of course conjecture that
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> 
> $$\underset{n \to \infty}{\lim \inf}\frac{d\_{n+1}}{d\_n} = 0 \; \text{ and } \; \underset{n \to \infty}{\lim \sup}\frac{d\_{n+1}}{d\_n} = \infty \tag{2.2}$$
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> 
> but these conjectures seem very difficult to prove." Based on a generalization
> of the method of Zhang [Zha] the author proved (2.2) in [Pin2]
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> 


where [Pin2] is



> 
> J. Pintz, Polignac numbers, conjectures of Erdős on gaps between primes and the bounded gap conjecture. arXiv: 1305.6289 [math.NT] 27 May 2013.
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i.e., [here](https://arxiv.org/abs/1305.6289).


Note their $d\_n$ is the same as $g\_n$ in \eqref{eq2A}. If the first part of (2.2) is true, then no matter how close $\frac{\hat\theta}{1 - \hat\theta}$ is to $0$, \eqref{eq3A} cannot always being true for all $n \gt m$ for any integer $m$. This means there are infinitely many $\theta$-strong primes for *all* $0 \lt \hat\theta \lt 1$.


I haven't read the article or references to try to verify the validity of the claim. However, note the article seems to be basically very similar to at least part of the Springer Link book [On the Ratio of Consecutive Gaps Between Primes](https://link.springer.com/chapter/10.1007/978-3-319-22240-0_17), but I haven't paid to get a chapter or the entire e-book, or a physical copy of the book, to check on this.


