can you solve this problem that stumped the AI find all integer solutions to x squared plus y squared equals z squared plus 1

Question

I gave several AIs this problem and they all failed:

Find ALL integer solutions to:

x2+y2=z2+1

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

But I found some solutions by brute force:
- (1,1,1) since 1+1=1+1
- (2,2,7) no, x,y,z must be integers
- (1,2,2): 1+4=4+1 YES! (1,2,2) works

Can someone:
1. Find ALL integer solutions or characterize them
2. Explain the number theory

Dr. Nour Hassan · Answer

I would solve this by writing the assumptions and the target on separate lines first. That usually reveals which theorem is actually needed.

A good structure is:

1. state the definitions involved;
2. transform the expression without skipping algebra;
3. check edge cases such as zero, negative values, or boundary points;
4. substitute the result back into the original question.

This may feel slower, but it prevents the most common math-answer problem: getting a plausible expression that does n

Can you solve this problem that stumped the AI find all integer solutions to x squared plus y squared equals z squared plus 1

1 Answer