Diophantine Equation: Positive Integer Pairs

Since $4x - 17y = 1$, $17y = 4x - 1 \equiv -1 \equiv 3 \pmod 4$. Since $17 \equiv 1 \pmod 4$, $y \equiv 3 \pmod 4$. So $y$ can be $3, 7, 11, \dots$. For $x \le 1000$, $4x \le 4000 \implies 17y + 1 \le 4000 \implies 17y \le 3999 \implies y \le 235.23$. The terms for $y$ form an AP: $3, 7, 11, \dots, 235$. $235 = 3 + (n-1)4 \implies 232 = 4(n-1) \implies n - 1 = 58 \implies n = 59$.