Wouldn't you want to search for numbers divisible by 4? I must be missing something here.
I wanted 8 or 16 for my third guess to add one or two more factors of 2 beyond just being divisible by 4. That way, if my next clue still said divisible by 4, I'd eliminate around half the candidates. If it said 8, but my guess was divisible by 16, that would divide the candidates by 4.
Hey, same here! Today I tried (499, 744). 499 gives "largest prime divisor is 7", and 744 gives "x-744 is a square".
You made this more complicated than necessary given that you involved Wolfram Alpha. I also started (499, 744). At that point, (x-744) had to be a square number that's divisible by 7. That left only 7² and 14² as 21² is too big. Unfortunately, neither 793 - 499 = 294 nor 940 - 499 = 441 have a prime factor larger than 7, which makes it a coinflip.