# Thread: ordered integer pair,s (x,y)

1. ## ordered integer pair,s (x,y)

(1) If $(1!)^2+(2!)^2+(3!)^2+.......................+(x!)^ 2 = y^2$. Then ordered integer pair,s of $(x,y)$ , where $(x,y)\in \mathbb{N}$

(2) If $(1!)^3+(2!)^3+(3!)^3+.......................+(x!)^ 3 = y^3$. Then ordered integer pair,s of $(x,y)$ , where $(x,y)\in \mathbb{N}$

2. ## Re: ordered integer pair,s (x,y)

(1,1) is one such ordered pair for both of those. I ran a check in Mathematica, and up to x=1000, there are no other valid pairs for either of them. You might try continued fractions...

$\cfrac{\cfrac{\cfrac{y^2-1}{2^2}-1}{\cdots}-1}{x^2} = 1$ (for part (1))

$\cfrac{\cfrac{\cfrac{y^3-1}{2^3}-1}{\cdots}-1}{x^3} = 1$ (for part (2))