Let x, $y$ be positive integers such that $x$ 4 = ( $x$ − 1)(y 3 − 23) − 1 . Find the maximum possible value of $x$ + y.

Step 1: 4n + $r$ = $k$ in I

Step 2: => 4n + $r$ = $k$ 2

Step 3: ( $x$ 4 + 1)

Step 4: = $y$ 3 − 23 , x, $y$ in I

Step 5: Since, $x$ \neq 1 => x2

Step 6: y 3 − 23 in I

Step 7: => inI

Step 8: Let, $x$ – 1 = p

Step 9: (1 + $p$ )4 + 1

Step 10: (a4 $p$ 4 + a3 p3 + \dots..a1p + 2)

Step 11: => => $p$ divides 2

Step 12: => $p$ = {–2, –1, 1, 2}

Step 13: => $x$ in {–1, 0, 2, 3}

Step 14: but $x$  2 => $x$ = 2 or 3

Step 15: If $x$ = 2 24 = 1 × (y3 – 23) – 1

Step 16: => (17 + 23) = y3

Step 17: => $y$ I

Step 18: If $x$ = 3 (34 + 1) = 2(y3 – 23) = $y$ = 4

Step 19: => x+y=7