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The longer I study mathematics, the more I think that we actually don’t know that much. I mean of course we don’t know that much, because theoretically mathematics is an infinitely big field of research and we as a species supposedly will only live for a finite time. So it would be reasonable to say that we don‘t know anything in mathematics. But that’s a little bit too philosophical for my question.

Now I always wondered how some mathematicians even came up with theorems that are really hard to prove. But now I think it is really easy to come up with a conjecture. The only difference would be that this conjecture probably won’t have many interesting consequences.

So what do I mean by that?
Let me state the following:

Let $p$ be a prime number. $p$ is called the $n$-th counting prime, if the sequence $123…n$ is somewhere in it’s decimal representation. So $11$ would be the first counting prime and $127$ the second counting prime.

My conjecture would now be: for any $n \in \mathbb{N}$ the $n-th$ counting prime number does exist.

So this just came to my mind and I think it would be an open problem. If it is then I would like to call it the Matthias Conjecture.

And in here you could replace prime numbers with powers of two, three, seven or the round off of the imaginary part of non-trivial solutions to $\zeta(s) = 0$, where $\zeta$ is the Riemann zeta function…. Or even the real part of these solutions :)

I am a little bit frustrated with this thought, because for the most part I had the feeling that we can do so much with mathematics today, but if even I could state something which is unknown and hard to prove, I rather feel like we really don‘t know anything…
So what do you think?

I don’t really know if this fits in with this site; if not I can delete it.

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    $\begingroup$ It does exist by looking at the prime number theorem. $\endgroup$ Commented Sep 26, 2024 at 6:03
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    $\begingroup$ @Tre, or Dirichlet's Theorem on primes in arithmetic progression (which came sixty years before the prime number theorem). $\endgroup$ Commented Sep 26, 2024 at 8:54
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    $\begingroup$ I agree that it's easy to state problems that are hard to solve (but I don't think you have done a good job of it). $\endgroup$ Commented Sep 26, 2024 at 8:56
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    $\begingroup$ It seems wrong to use the phrase "the $n^{\text{th}}$ counting prime", since primes with the given conditions are not unique. I'd say "an $n^{\text{th}}$ counting prime", or add something like "...and the least such prime" at the end of the definition. $\endgroup$ Commented Sep 26, 2024 at 16:50
  • $\begingroup$ I think the goal of mathematicians is not to find problems that are "hard to prove", but problems that are interesting (which is fully subjective). Or better, they are interested in finding structures and relations between mathematical objects. If some theory easily explains connections between different domains, they will be happy (and then move on to something else, because the life of a mathematician is a never ending task) $\endgroup$ Commented Sep 27, 2024 at 4:34

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You can't really say that you have created "many" problems simply by replacing with "prime numbers with powers of two, three, seven or the round off of ..." because a single technique may solve all of them in one fell swoop. In this case, we can fix the digits to appear at the front of the prime, and analyzing the density of primes immediately gives you the result. This marks the beginning of analytic number theory.

It is true that there are many problems stated in elementary language that are currently unreachable. And it is true that there are infinitely many problems that will provably never be "solved". (Of course you may consider a proof of unsolvability to be a solution itself.) But if something cannot be proved or disproved by stupid reasons, such as immediately finding a counterexample, then we can hope to find some beautiful theory in the pursuit — It's possible for something to hold by coincidence for a few cases, but things like Fermat's last theorem holds for infinitely many cases, in fact all of the cases except $n = 1,2$, so we have a reason to believe there is some deep structure at work.

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There are 3-4 Points to look into here :
POINT A : OP : I think it is really easy to come up with a conjecture
POINT B : OP : Is it actually really easy to state conjectures, which are open and on the first view really hard to prove?
POINT C : OP : My conjecture would now be, that for all $n \in N$ , the $n^{th}$ counting prime number does exist.
POINT D : PREM : I make a Conjecture ( Prem Pi Prime Conjecture ) in about 5 minutes , to check where it goes

POINT C : We can easily show that your Conjecture is true.
Let us write the number $X=123 \cdots n$ , which might be Odd or Even , hence divisible by 2. We can append a Digit $1,3,7,9$ , I will add $1379$ to make the number $Y=123 \cdots n1379$ which is always Odd.
Let that have $m$ Digits. Then $Z=10^{2m}$ has a lot so Zeros.
Consider $Z+Y$ , $2Z+Y$ , $3Z+Y$ , $4Z+Y$ , $5Z+Y$ , ... , $nZ+Y$ Due to a well known theorem ( Dirichlet ) , this Arithmetic Sequence will have Primes occurring in it.
Naturally , that Prime has the "Wanted Sequence" in it , hence the $n^{th}$ counting prime number does exist.
Conjecture was easy to make , Proof was easy too.
All your variations are covered too.

POINT A : Yes , it is easy to make Wild Conjectures , even non mathematicians can do it.
Every body can try Chess , even non grandmasters , though the games involving two learners will not get into news or history books or chess record books.
Every body can make up stories , not every body can write like Shakespeare.
Every body can sing , not every body can sing like Shakira.

POINT B : Yes , it might be actually really easy to state conjectures, which are open and on the first view really hard to prove , though we have to include some more criteria there :
B1 : It must be the outcome of deep thinking and analysis , not armchair random thinking , thrown out the next evening. B2 : It must have strong evidence to support it , not just guess work. We can not exclude trying to get the evidence just because it requires hard work.
B3 : It must have wide impact.
B4 : It must be interesting.
B5 : It must resist "Proof Attempts" over a long time.
B6 : There must be certain reasons to believe that it must be true.

Naturally , two learners making Wild Conjectures might not get into news or history books or math text books.

POINT D : I claim that the Decimal Digits of $\pi$ contain all Prime numbers. Eg $\pi=3.141\underline{\color{violet}{5}}9\underline{\color{violet}{2}}65\underline{\color{violet}{3}}589\underline{\color{violet}{7}}932384626433832795$
Here I have highlighted $\color{violet}{2,3,5,7}$ , though I can not yet see $11,13,17,19$

I call this Prem Pi Prime Conjecture.
We can easily see that Proof is going to be hard using currently known techniques.
It has no impact. It might be a curiosity for a while , to be forgotten shortly.
It is not the outcome of deep thinking.
I have not collected evidence to check whether all Primes till 1 million do occur or not : It is just too much hardwork ! Let me leave it like a Conjecture , making somebody else to work on it !
It is generally taken ( without Proof ! ) that $\pi$ is "normal" [ Details ] , which means that all Digits will occur. It is then almost a Corollary that all Primes might occur.

Will the Math Community accept that Conjecture ? I think not ! I have to put in more work over a long time to justify it !

SUMMARY :

It is easy to make Conjectures with no effort , no thinking , no evidence , no impact.
It is not easy to make Conjectures according to the "Accepted Standards" in the Modern Era.

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    $\begingroup$ Great Answer thanks:). I wasn‘t familiar with that dirichlet theorem. But also an quick implication is that any arbitrary finite sequence of numbers can be found in some prime right? Then one could say something like „Everything, that will be known is already in some prime number“. That sounds neat. Thanks again. $\endgroup$ Commented Sep 26, 2024 at 14:09
  • $\begingroup$ That quick implication is true , we have to make suitable arithmetic sequences. I am not sure what you mean by every thing known , though that might be valid too. We could alternately claim that all sequences might occur in Pi Digits which is almost a corollary of the normality. $\endgroup$ Commented Sep 26, 2024 at 14:56
  • $\begingroup$ Just as an aside, 11, 13, 17 and 19 appear within the first 160 digits of $\pi$. $\endgroup$ Commented Sep 26, 2024 at 22:17
  • $\begingroup$ I mean everything that will be known can obviously be written down on a finite amount of paper. And if something can be written down it could also be expressed through a number(like a binary code or something). This string of numbers have always been in some prime number. I know that‘s really more on the philosophical site, but math is just applied philosophy, isn‘t it? $\endgroup$ Commented Sep 26, 2024 at 22:22
  • $\begingroup$ Valid Observation , @MathMatthias , it is likely true. $\endgroup$ Commented Sep 27, 2024 at 5:14
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Any finite amount of mathematical knowledge (such as all current mankind mathematical production) is zero compared to what could be known, and will ever be.

The accumulated mathematical knowledge is huge, and no single man could have even a general grasp of the available material. The list below is probably incomplete and superficial: https://en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics. By the way, most of modern mathematics is extremely difficult and only accessibly to professionals.

Specialists in their respective domains are probably able to tell an interesting conjecture from a statement of little value. For lambda amateurs, about all that was upon their reach has been addressed, except maybe in recreational mathematics.

Despite these difficulties, mathematics are extraordinarily useful.

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This will sound tautological, but what you know is defined by what you know, not what you don't know. The unlimitedness of mathematics does not mean that some finite knowledge is nothing.

As for the construction you describe, I can come up with an even more brutal version using cryptography. Consider the application of cryptographic hash functions like AES. Consider:

Cort's Conjecture N/X (where N is any integer and X is any 256-bit number): The first AES-256 preimage of X which starts with N when treated as a number, is odd.

Using that pattern, I can construct an unbounded number of conjectures, without even trying. I just vary N or X. So in that sense, there's a lot we don't know.

On the other hand, there's a really interesting conjecture that identifying preimages of AES-256 is hard. A whole slew of really useful and applicable cryptographic results from this conjecture flow forth and make AES a very important algorithm to this day.

So I'd argue that number of conjectures is not the best tool to discuss what we know or don't know. The whole beauty of some systems, such as cryptographic algorithms, is in the fact that it's so hard to know things. The conjecture that it is hard to make a preimage attack on AES is more profound and more useful than all of my unbounded set of conjectures put together.

... is it too late to take my name off of Cort's Conjecture N/X? I'm not really all that proud of them.

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