Can you solve this integer equation that stumped the AI?

Question

I gave several AIs this problem and they all failed:

Find ALL integer solutions to:

x^2 + y^2 = z^2 + 1

The AIs kept saying "Pythagorean triples" and suggesting $(3,4,5)$ doesnt work because $9 + 16 = 25 \neq 26$ .

But I found some solutions by brute force:

$(1, 1, 1)$ since $1 + 1 = 1 + 1$
$(2, 2, \sqrt{7})$ no, $x, y, z$ must be integers
$(1, 2, 2)$ : $1 + 4 = 4 + 1$ YES! $(1, 2, 2)$ works

Can someone:

Find ALL integer solutions or characterize them
Explain the number theory behind this equation
Show why AI struggled with this while a human approach works

Chen Wei · Accepted Answer

The equation is x 2 + y 2 = z 2 + 1. It has many integer solutions, so the goal is not to prove there are none. A simple family comes from setting y = 1: x 2 + 1 = z 2 + 1 , so x 2 = z 2 . That gives x = ± z . So every triple ( x , y , z ) = ( n , 1 , n ) or ( x , y , z ) = ( − n , 1 , n ) is a solution. There are also nontrivial examples, like 3 2 + 5 2 = 34 and 33 is not an integer, so not every pair works. The useful lesson is that an AI answer claiming a neat finite list is suspicious unless it proves why all other cases are impossible.

Can you solve this integer equation that stumped the AI?

1 Answer

Can you solve this integer equation that stumped the AI?

1 Answer