Suppose you want to factorize 35.  √35 is a little less than 6.  So there might be (6-a)(6+a) that equals 35.  Or if not, maybe (7-a)(7+a) or if not, maybe (8-a)(8+a).   (6-1)(6+1) = 5*7 = 35.  Success.

Now try 99899 whose square root is a bit over 316. 

Maybe (317-a)*(317+a) or maybe (318-a)*(318+a).

317^2 - 99899 = 590

318^2 - 99899 = 1225  which is a perfect square (35^2), bingo.

(318-35)(318+35)

283 * 353

I haven't proved 283 and 353 are prime, however.