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.
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.