6

Many modern proofs/conjectures concerning prime numbers (twin primes, infinite primes not containing one specific digit) intend to show that some property of prime numbers is infinite.

Are there any similar, nontrivial properties of prime numbers that have been proven to only appear within the first n, but not the rest (finite)? As a rough guideline to nontriviality, there should be at least ~3 somewhat spread out instances of the property that discontinue after reaching some number above ~50. The property cannot solely involve a simple operator such as < or >, and ideally should not involve repeated deletion of digits (truncatable primes) as this makes proof via exhaustion easier.

dcsuka
  • 209
  • 2
    Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Nov 26 '22 at 00:09
  • 1
    The primes less than $1000$ applies to only the first $n$ primes for some $n.$ I mean, this question is way too broad. – Thomas Andrews Nov 26 '22 at 00:13
  • Sorry, I am having trouble phrasing my question in such a way that it gets a nontrivial answer. I am looking for a type of property that is similar in nature to the twin primes conjecture or the other one I mentioned, even though I cannot find a specific way to express that. – dcsuka Nov 26 '22 at 00:17
  • 2
    @Dan your intention is clear, but I can assure you, the first $n$ primes are not less than or equal to $n$. – Clayton Nov 26 '22 at 00:23
  • @Dan The first three primes are not less than or equal to $3$. – jjagmath Nov 26 '22 at 00:24
  • 1
    For some interesting examples, see Conjectures that have been disproved with extremely large counterexamples – Jair Taylor Nov 26 '22 at 00:47
  • The question is actually interesting, therefore I upvoted , but it definitely needs more focus. That an example like "the primes less than $1000$ does not quality as nontrivial" should be obvious, but where is the borderline ? – Peter Nov 26 '22 at 10:34
  • I am thinking of quadratic polynomials that can generate primes but it was proven that no quadratic polynomial can generate all the primes. – user25406 Nov 26 '22 at 12:59
  • @Peter I am open to suggestions/edits about how to make the criteria for nontriviality more concrete and rigorous. Until now, I have just continually updated the criteria upon hearing more examples that seem trivial. – dcsuka Nov 26 '22 at 16:04
  • Unfortunately we are beholden to you, the asker, to compose a problem that can be answered by reasoned mathematical arguments. Several edits to update criteria "upon hearing more examples that seem trivial" will strike many as moving the goalposts. I would note that you have posted an Answer, which is perhaps one way of staking out the results you would accept as non-trivial. Closely related. – hardmath Nov 26 '22 at 19:10

3 Answers3

5

While this assertion is still a conjecture rather than a proven statement, it's otherwise a classic example of exactly what the OP is asking:

We believe that there are only finitely many Fermat primes, that is, primes that are one more than a power of 2. Indeed, we believe that the only Fermat primes are 3, 5, 17, 257, and 65537.

(As a consequence, there would be only finitely many odd numbers $n$ such that a regular $n$-gon can be constructed with a ruler and compass.)

Greg Martin
  • 78,820
4

Some possible solutions after digging into this problem a bit deeper:

  • Primes where you can repeatedly delete any number and still have a prime: 2, 3, 5, 7, 23, 37, 53 and 73

  • Left truncatable primes

  • Right truncatable primes

  • Left and right truncatable primes

https://en.wikipedia.org/wiki/Truncatable_prime

However, all of these involve some sort of deletion of numbers, which make proof via exhaustion easy. The most interesting solution would not involve deletion.

EDIT with sufficient answer:

Upon consulting Wikipedia's list of prime numbers, it appears a few more candidate sequences exist including supersingular primes, minimal primes, Wilson primes, Wolstenholme primes, Fermat primes, and Stern primes.

dcsuka
  • 209
  • (+1) because at least we are dealing with proven results and that only finite many deletable primes exist is not obvious. Seems to be a proper example of what the author wants. – Peter Nov 26 '22 at 10:36
1

There are some easy examples, as:

The prime $2$ is the only even prime.

The only primes that are consecutive integers are $2$ and $3$.

The only triple of primes differing by $2$ (that is: $p$, $p+2$ and $p+4$ are primes) is $(3,5,7)$.

The only prime of the form $n^4+4$ is $5$.

These are very elementary examples. I think the question is interesting. It would be nice to have a less trivial example of a theorem that is valid only for a finite number of primes.

jjagmath
  • 18,214
  • Come to mind: bounds relating $n$ and $p_n$ based on the PNT, imaginary quadratic fields with class number $1$ – reuns Nov 26 '22 at 00:39
  • Thank you very much for your answer. Seeing these examples has helped me try and refine the criteria some more, to make the idea of nontriviality more concrete. Please edit if you can think of any further guidelines. – dcsuka Nov 26 '22 at 00:50
  • 3
    The only primes $p$ such that $n^2+n+p$ is simultaneously prime for all $0 \le n < p-1$ are $2,3,5,11,17,41$. (See https://en.wikipedia.org/wiki/Heegner_numbe.) – Erick Wong Nov 26 '22 at 00:59
  • 2
    @ErickWong Link corrected to Heegner Number. – user2661923 Nov 26 '22 at 01:03
  • I do not think that those example are interesting, they are too trivial. Covers for numbers of the form $2^n+k$ with given $k$ showing that this can never be prime could come nearere to what the author actually wants. – Peter Nov 26 '22 at 10:30