Why is $3p -10$ where $p$ is a prime, equal another prime or semiprime?

Question

I was wondering if this equation had any implications for prime research: $$3p-10$$ where $p$ is a prime.

I found this interesting and was curious if it had any backing through any prime proofs.

Here is my python script that will do the number crunching: https://github.com/Storms-Engineering/Prime-Scripts

Sample of output:

Skipped:5
Skipped:7
Skipped:11
Skipped:13
Skipped:17
Skipped:19
Skipped:23
Nope: 29
Output: 77
Factored: [7, 11]
Skipped:31
Skipped:37
Skipped:41
Nope: 43
Output: 119
Factored: [7, 17]
Skipped:47
Skipped:53
Skipped:59
Skipped:61
Skipped:67

Any thoughts?

Also if there is a better place to discuss this, I am all ears. This was really the only place that I could think of to ask this.

Answer 1

By the way, here is an "all-you-can-eat" approach to generate counterexamples. Find a series of prime $7 \leq p_1 < \cdots < p_l$, and let $N = p_1^{e_1} \cdots p_l^{e_l}$. $10$ is relatively prime to $N$, so there is a number $m$ such that $3m \equiv 10 \pmod N$.

Find a prime $x$ such that $x \equiv m \pmod N$. This is always possible by Dirichlet's Theorem on Primes in Arithmetic Progressions.

Then, $N \ | \ 3x -10$, which is very composite!

Answer 2

For $3p-10>0$, it must be true that $p>3$. All primes greater than $3$ have the form $6n\pm 1$.

So $3p-10=3(6n\pm 1)-10=\{18n-7,18n-13\}$, either of which has the form $6n'-1$. Thus all numbers of the form $3p-10$ have a form that might possibly be a prime. For small numbers of the form $6n'-1$, the significant majority are prime: $5,11,17,23,29,41,47,53,59,71,83,89,101,107$ vs $35,65,77,95$, so more or less by coincidence, you find a lot of hits in the first (smaller) examples you look at, and it seems like there might be some kind of pattern. But if you look at enough larger examples of $p$, you will find plenty that do not produce a prime by this formula.