Simple trick: divide by 100, 10000 or 1000000 use Newtons method on the integers, then multiply by 10,100 or 1000 and add a 'fudge factor' based on how large the fraction was...
It's cheating but it can get you pretty close, I'd peg it at 15240 using that trick. If you just want to do the closest squares you can average between 15 (225, too low) and 16 (256, too high) so you'd guess 15500, which is much too high, but one more iteration of Newtons method gets you closer than what chat gpt gives. You can already see that because 225 is much closer than 256 and that puts you closer to 15250 than 15500. And 15250 is actually not a bad guess at all.
And if chatGPT said “I don’t know the actual answer but my best guys is 15229” that would be a reasonable and potentially useful answer.
The fact that it gives you a number that isn’t rounded to the nearest tens, hundreds, or thousands place means that it doesn’t look like an approximation to any reasonable person, which makes it a terrible answer.
My younger brother used to have this problem. If you asked him a question like “how long until you get here”, he’d say “17 minutes”. What he really meant was “around 20 minutes”, but everyone thought he must know the exact time. Like he’d done the drive many times, or he was looking at his GPS.
Simple trick: divide by 100, 10000 or 1000000 use Newtons method on the integers, then multiply by 10,100 or 1000 and add a 'fudge factor' based on how large the fraction was...
It's cheating but it can get you pretty close, I'd peg it at 15240 using that trick. If you just want to do the closest squares you can average between 15 (225, too low) and 16 (256, too high) so you'd guess 15500, which is much too high, but one more iteration of Newtons method gets you closer than what chat gpt gives. You can already see that because 225 is much closer than 256 and that puts you closer to 15250 than 15500. And 15250 is actually not a bad guess at all.