Take $n = 2039183$ . The square root is approximately 1428, so that's what we set our iterator's initial state to. The test cap is 339865.

At $i = 1428$ , we find that $\sqrt{1428^2 - 2039183} = 1$ , clearly an integer.

Then, $1428 - 1 = 1427$ and $1428 + 1 = 1429$ . By multiplication, we verify that $2039183 = 1427 \cdot 1429$ , indeed. It is the product of a twin prime. By trial division, this would have taken 225 test divisions.

However, there are integers for which trial division performs much better than the Fermat method. For example, take $n = 1411041$ . Our iterator starts at 1188 and the test cap is 235175.

At $i = 1188$ , we find that $\sqrt{1188^2 - 1411041} \approx 17.4068952$ . Similar frustration at 1189, 1190, etc.

It's not until we get to $i = 235175$ that we finally get $\sqrt{235175^2 - 1411041} = 235172$ .

Then, $235175 - 235172 = 3$ , and $235175 + 235172 = 470347$ , and we verify that indeed $3 \cdot 470347 = 2039183$ . This took 233987 instances of squaring, subtraction and square root extraction, which would have taken just 196 test divisions in trial division.

How about 4393547637856664251490043044051018234292171475232959? The square root is approximately 6628384145368058140794809 and the test cap is 732257939642777375248340507341836372382028579205495. At worst, the Fermat method could take as many as 732257939642777375248340500713452227013970438410686 instances of squaring, subtraction and square root extraction to give a result. Clearly a more sophisticated method is needed to factorize an integer of this magnitude.