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-by-step solution

aligned & 4 n+r =k I & 4 n+r=k^2 & (x^4+1 ) (x-1) =y^3-23, x, y I aligned Since, x 1 aligned & x 2 & y^3-23 I aligned x^4+1 x-1 I Let, x-1=p aligned & (1+p)^4+1 p & (a₄ p^4+a₃ p^3+ . . a₁ p+2 ) p aligned array ll & 2 p p divides 2 & p= -2,-1,1,2 & x -1,0,2,3 array but x 2 x=2 or 3 aligned & If x=2 2^4=1 (y^3-23 )-1 & (17+23)=y^3 & y I aligned If x=3 (3^4+1 )=2 (y^3-23 )=y=4 x+y=7