Solution of Ljunggren Equation

Theorem
The indeterminate Diophantine equation:
 * $x^2 + 1 = 2 y^4$

has the single solution:
 * $x = 239$
 * $y = 14$

Proof
Setting $x = 239$:

and so $y = 13$.

Also see

 * Largest Prime Factor of $n^2 + 1$