Examples of Theorems of the form "If a set could contain a prime, it will contain a prime".

Question

One of my favourite theorems in mathematics is Dirichlet's theorem on arithmetic progressions, which states that every set of arithmetic progressions $\{a, a+d, a+2d, \dots \}$ will contain a prime as long it is not trivial that it won't, which is, as long as $a$ and $d$ are coprime.

The behaviour that "every set that could contain a prime will contain a prime" seems to be in the nature of prime numbers, but there are many examples where statements like this are mere conjectures, not proved theorems. Some examples include the twin prime conjecture or the problem of the existance of infinitely many Mersenne-primes.

I wonder, are there more actual theorems of this form?

There are infinitely many primes that start with any specified string of digits. See, e.g., this. — lulu, Jun 02 '21 at 15:41
Artin's Conjecture on primitive roots seems like another example — Rushabh Mehta, Jun 02 '21 at 15:51
To add another example, Huxley's theorem on primes in short intervals. Every interval of the form $[x,x+x^\theta]$ with $\theta > \frac{7}{12}$ contains a prime. On the Riemann Hypothesis, one can replace $ \frac{7}{12}$ with $\frac{1}{2}$. — Joshua Stucky, Jun 02 '21 at 18:07

score 1 · Answer 1 · answered Jun 02 '21 at 17:04

1

Yes, there is another example: Piatetsky Shapiro primes.

There is a $c>1$ for which infinitely many numbers of the form $\lfloor k^c\rfloor$ are prime.
Check here for details.

answered Jun 02 '21 at 17:04

Konstantinos Gaitanas

9,115

Examples of Theorems of the form "If a set could contain a prime, it will contain a prime".

1 Answers1