Which prime $p$ makes $\frac{7^{p-1}-1}{p}$ and $\frac{11^{p-1}-1}{p}$ be perfect squares? (not simultaneously) Let $p$ be a prime number. Then which $p$ makes
$$\frac{7^{p-1}-1}{p}$$
be a perfect square? Similarly, which $p$ makes
$$\frac{11^{p-1}-1}{p}$$
be a perfect square?
 A: This is not quite Ramanujan-Nagell or Nagell-Ljunggren, there may not be anything published. So, generalize. 
One thing to play with is this: you have chosen an odd prime, so $p-1$ is even, and the power of $7$ or $11$ is a square; a very specific square, but still. Writing this as $x^2,$ we are asking when $x$ is a prime or prime power when 
$$   x^2 - p y^2 = 1.$$
For completeness, given the trivial solution $(1,0)$ to  $   x^2 - p y^2 = 1,$ we get an infinite sequence of (all positive) solutions by Pell automorphisms,
$$ p=2: \; \; \; (x,y) \mapsto (3x+4y,2x+3y);    $$
$$ p=3: \; \; \; (x,y) \mapsto (2x+3y,x+2y);    $$
$$ p=5: \; \; \; (x,y) \mapsto (9x+20y,4x+9y);    $$
$$ p=7: \; \; \; (x,y) \mapsto (8x+21y,3x+8y);    $$
$$ p=11: \; \; \; (x,y) \mapsto (10x+33y,3x+10y);    $$
$$ p=13: \; \; \; (x,y) \mapsto (649x+2340y,180x+649y);    $$
$$ p=17: \; \; \; (x,y) \mapsto (33x+136y,8x+33y);    $$
$$ p=19: \; \; \; (x,y) \mapsto (170x+741y,39x+170y).    $$
The evident 2 by 2 matrix that accomplishes this when multiplying a column vector is
$$   
 \left(  \begin{array}{rr}
  A  &  pB  \\
   B   &  A  
\end{array} 
  \right)  ,
  $$
which must have determinant $1.$
A way to pass the time, anyway. 
