Any idea?

Well, this follows immediately from Big Spoiler

hmm i did not know about that theorem. can someone give me details about this

Sure. This is part 1 of our solution https://calimath.org/pdf/IMOC2023-N5-(part-1).pdf. So, to finish the proof, you want to show that hint.

Since we decided to post our solution in two separate videos on our Youtube channel (part 1: https://youtu.be/YSbrN3aOPYU), I will post the full proof next Friday.

Here is the full solution https://calimath.org/pdf/IMOC2023-N5.pdf and part 2 on our youtube channel https://youtu.be/YSbrN3aOPYU.

Here are some remarks regarding the general theme underlying this problem: 1) As the solution in #5 and #6 shows, there is a relation between $x$ and the number of solutions in $\mathbb{F}_p^2$ of the congruence $u^2-t^4 \equiv 1 \pmod{p}$, namely the number of solutions is $p-1-x$. 2) Adding the two points at infinity, the number of solutions is $n=p+1-x$. 3) In fact, the variety defined by the equation $u^2=1+t^4$ is "essentially" an elliptic curve (more precisely it can be transformed birationally into an elliptic curve, i.e. an equation $y^2=g(x)$ for some cubic polynomial $g$). The general procedure for such a transformation is explained here. 4) For any elliptic curve over $\mathbb{F}_p$, there is the famous theorem of Hasse that the number of solutions (including points at infinity) is $n=p+1-x$ for some $\vert x \vert \le 2\sqrt{p}$. This is of course exactly the bound that we are trying to prove. 5) However, the proof in #5 and #6 was much more elementary than the proof of Hasse's theorem in the general case. Indeed, it solved the problem by proving that $x$ is the unique (up to sign) integer with $x=2a$ where $p=a^2+4b^2$ (by the way, reproving Fermat's theorem on sums of two squares). 6) This special phenomenon of "being able to evaluate $x$ explicitly" can be explained by the fact that the elliptic curve mentioned in 2) has a special property, called complex multiplication (roughly speaking this means that it has an exceptionally large set of symmetries). 7) For elliptic curves with complex multiplication, Hasse's theorem is elementary and was (essentially, but of course in a different language) already known to Gauß. Indeed, one can then always express $x$ "explicitly" in a way similar to 5). For another rather famous special case of this theorem, one can mention the result of Gauß that for $p \equiv 1 \pmod{3}$ and the elliptic curve $x^3+y^3+z^3=0$, the number of solutions is $p+1+A$ where $4p=A^2+27B^2$. And in fact, as mentioned here, one can again relate this to a certain binomial congruence, namely $A \equiv \pm \binom{4k}{2k} \pmod{p}$ where we write $p=6k+1$.