What are the total number of positive integral solutions of the equation
$x^4-y^4=3789108$
I literally have no clue how to conclude since there could be as many as infinite such numbers or there could be no such number. Please provide a mathematical logic to this question.
(Expanding on @Hari Shankar's comment)
Consider $x^4 - y^4 = 3789108$
Since the RHS is even, we have that $x,y$ are both even or both odd (same parity).
Suppose $x, y$ are both even. This implies $x = 2m$ and $y = 2n$. Hence $x^4 - y^4 = 16 (m^4 - n^4)$ which implies that $16\, |\, 3789108$ which is clearly incorrect.
Suppose $x,y$ are odd. This implies $x = 2m + 1$ and $y = 2n+1$. Then
$$x^4 - y^4 = (16m^4 + 32m^3 + 24m^2 + 8m + 1) - (16n^4 + 32n^3 + 24n^2 + 8n + 1) = 8 \cdot k$$
Or $8 \,|\,3789108$ which is clearly incorrect. Hence there are no solutions.