Is there a property in $\mathbb{N}$ that we know some number must satisfy but don't know which one?

Question

I have two questions.

$(1.)$ Is there a property of the natural numbers such that we know at least one number satisfies it but we don't know which one?

Even more,

$(2.)$ Is there a property of the natural numbers such that we know at least one number satisfies but we don't know which one and moreover we cannot bound any such number by another known number?

Now the question might arise of what does it mean to know a number. To be able to write it down in finite time given an infinite amount ink and paper would suffice.

Answer 1

Goldbach's conjecture is the statement that, if $n$ is an even natural number and $n>2$ then $n$ is the sum of two primes. It is not known whether this conjecture is true, nor is any bound known on how big the first counterexample could be if the conjecture is false. In view of this situation, consider the following property of a natural number $n$: Either $n$ is the first counterexample to Goldbach's conjecture or Goldbach's conjecture is true and $n=0$. We know that there is exactly one $n$ with this property (because either Goldbach's conjecture is true or it has exactly one first counterexample), but we don't have any reasonable bound for how big $n$ could be. (The weasely word "reasonable" is there to exclude "bounds" like "$7+$ the first counterexample to Goldbach's conjecture or $13$ if Goldbach's conjecture is true".)

Answer 2

For the first question, we know there is a smallest natural number $n$ such that $\pi(n)>\text{li}(n)$ where $\pi(x)$ is the prime counting function and li$(x)$ is the logarithmic integral, but we don't know what the number is.

EDIT: For the sake of having a more complete answer, I will include our discussion in the comments.

There has been much progress in determining an upper bound for the first integer at which this switch occurs, but still the number eludes us. Strangely enough, we know that there are infinitely many numbers for which this inequality holds.

Answer 3

Here's an interesting example that is the subject of recent (and celebrated) research:

Consider the set $\{p_1, p_2, p_3, \ldots\}$ of prime numbers, listed in increasing order. The $n$th prime gap is the difference $p_{n + 1} - p_n$. In 2013, Yitang Zhang showed that there are infinitely many prime gaps of size $\leq 7 \cdot 10^7$. Since there are only finitely many positive integers smaller than this size, at least one of them must occur infinitely often, but we don't know which. Since then, the Polymath project has improved that upper bound to $246$ (as of last December). The famous and longstanding Twin Prime Conjecture says that $2$ occurs infinitely often.

Answer 4

It is known that one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational, but no one knows which one:

W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. doi:10.1070/RM2001v056n04ABEH000427.

Answer 5

Consider $\Sigma(m,n)$ to be the busy beaver function with $m$ states and $n$ colors.

The following is then very hard to find:

What is the smallest $m$ such that $\Sigma(m,2)>\Sigma(10,3)$?

Since there is a color reduction method for reducing a $k$ state, 3 color machine to a $7k$ state, 2 color machine, it is known that $m\leq 70$. On the other hand, it is clear that $m>10$. But it is very hard to say anything between this.

Answer 6

Since we do not know , for example $\Sigma(5)$ , we do not know which number $n$ satisfies $n=\Sigma(5)$, but we know that some number $n$ must satisfy the property. In this case we do not even have an upper bound.

Another less trivial example : The $20$-th Fermat-number $2^{2^{20}}+1$ is known to be composite, but no factor is known.

Denote $n$ to be the smallest prime factor of the $20$-th Fermat-number. This number is well-defined but we do not know it, but in this case, we can bound it.

If this does not hit your intention, please precise which kind of properties you mean.