Can $pk+1$ divide $(p-k)^2$?

Claim: If $a,b$ are positive integers and $q(a,b) := \frac{(a-b)^2}{ab+1}$ is an integer, it is a square number. This may be proved by Vieta jumping as follows:

Let $k \geqslant 1$ be an integer, which is not a square. Assume that there are positive integers $A,B$ such that $q(A,B) = k$, and choose $A$ and $B$ so that $A \geqslant B$ and that $A+B$ is minimal.

By expanding the statement $q(A,B) = k$, we find that $A$ is a solution to the quadratic $X^2 - B(2+k)X + B^2 - k = 0$. Since its discriminant is positive, there is another solution $C$ which may be expressed using Vieta's formulas as

\begin{align*} C = B(2+k) - A = \frac{B^2 - k}{A} < A. \end{align*} The first representation of $C$ shows that it's an integer, and the other one shows that it's non-zero, since $k$ isn't a square. Also, since \begin{align*} \frac{(C-B)^2 }{CB + 1} = k > 0, \end{align*} we see that $C$ has to be a positive integer. But then \begin{align*} C+ B < A+B, \end{align*} contradicting the minimality of $A+B$. $\blacksquare$

This means that if $q(p,t)$ is an integer, there is some integer $m$ with

\begin{align*} q(p,t) = \frac{(p-t)^2}{pt + 1} = m^2, \end{align*} and by estimating $q(p,t)$ (using $1 \leqslant t \leqslant p-1$) it's easy to see that $m^2 \leqslant p-1$. Rearranging this equation gives \begin{align*} p^2 - 2pt + t^2 = m^2 p t + m^2, \end{align*} so \begin{align*} p \mid p^2 - (2 + m^2) p t = m^2 - t^2 = (m-t)(m+t). \end{align*} Given the bounds on $m$ and $t$, the only possibilities here are that $m = t$ or that $m+t = p$. In the first case, our equation is \begin{align*} p^2 - 2pt + t^2 = t^3 p + t^2 &\Leftrightarrow p^2 - 2pt - t^3 p = 0 \\ &\Leftrightarrow p = 2t + t^3 \end{align*} so $t \mid p$, meaning $t = 1$, so $p = 3$, which is impossible. In the second case, we have $m = p-t$, so our equation is \begin{align*} (p-t)^2 = p^2 - 2pt + t^2 = (p-t)^2 p t + (p-t)^2 &\Leftrightarrow 1 = pt + 1, \end{align*} forcing $t = 0$, which is impossible.

Suppose that $p$ and $k$ are integers such that $p>k>0$ and $$n=\frac{(p-k)^2}{pk+1}$$ for some integer $n>0$. Note that $(x,y)=(p,k)$ is a solution to the quadratic equation $$x^2+y^2-(n+2)xy=n.\tag{1}$$ Let $(x,y)=(a,b)$ be a solution to (1) such that $a>b>0$ and $a+b$ takes the smallest possible value. We claim that $n=b^2$.

Observe that $$(x,y)=\big(b,(n+2)b-a\big)=\left(b,\frac{b^2-n}{a}\right)\tag{2}$$ is an integer solution to (1), but not necessarily positive. If $b^2>n$, then we see that (2) gives an integer solution in which $$b>\frac{b^2-n}{a}>0\text{ and }b+\frac{b^2-n}{a}<a+b.$$ This contradicts the assumption that $a+b$ is minimum. Hence, $b^2\leq n$.

If $b^2<n$, then observe that $$(x,y)=\left(\frac{b^2-n}{a},\frac{b^2-n}{a}(n+2)-b\right)=\left(\frac{b^2-n}{a},\frac{\left(\frac{b^2-n}{a}\right)^2-n}{b}\right)$$ is another integer solution to (1), with both numbers being negative. Since (1) is invariant under the assignment $(x,y)\mapsto (-y,-x)$, we see that $$(x,y)=\left(\frac{n-b^2}{a}(n+2)+b,\frac{n-b^2}{a}\right)$$ is a positive integer solution with $$\frac{n-b^2}{a}(n+2)+b>\frac{n-b^2}{a}>0.$$ By minimality of $a+b$, we must have $$\left(\frac{n-b^2}{a}(n+2)+b\right)+\frac{n-b^2}{a}\geq a+b.$$ Hence, $$n(n+3)\geq a^2+(n+3)b^2.$$ Recall from (2) that $$(n+2)b-a=\frac{b^2-a}{n}<0.$$ That is, $a>(n+2)b$ and we get $$\begin{align}n(n+3)&\geq a^2+(n+3)b^2>(n+2)^2b^2+(n+3)b^2\\&\geq (n+2)^2+(n+3)=n(n+3)+(2n+7)>n(n+3).\end{align}$$ This is a contradiction, so $n=b^2$ is the only possibility.

With $n=b^2$, (2) becomes $(x,y)=\big(b,(b^2+2)b-a\big)=(b,0)$, so $$a=b(b^2+2).$$ Note also that $$(x,y)=\big((b^2+2)p-k,p\big)=\left(\frac{p^2-n}{k},p\right)$$ is also an integer solution to (1) satisfying $x>y>0$, provided that $(x,y)=(p,k)$ is a solution to (1) with $p>k>0$. Indeed, we can see that all integer solutions $(x,y)=(p,k)$ with $n=b^2$ satisfying $p>k>0$ are given by $$(p,k)=(t_{j+1},t_j)$$ for some positive integer $t_j$. Here, we define $t_0=0$, $t_1=b$, and $$t_j=(b^2+2)t_{j-1}-t_{j-2}$$ for $j\geq 2$. Note that $t_j$ is divisible by $b$ at all $j$. Since $t_j>b$ for $j>1$, we conclude that $p$ is not prime when $b>1$.

For $b=1$, however, $t_j=F_{2j}$ (where $\left(F_m\right)$ is the Fibonacci sequence). Since $F_j$ divides $F_{2j}$ for all $j$ and $F_j>1$ for $j>2$, we conclude that $p$ is prime only when $p=F_4=3$.