 So let's use our quadratic factorization method to factor 539873. So our setup is to find numbers n, where n squared is a product of prime numbers less than or equal to some arbitrarily chosen b, which is the largest prime we want to work with. Since into this example we're working by hand, let's use prime numbers 19 or less, and so we'll be using 19 smooth numbers. So since square root of 539873 is approximately 734.76, we'll start with n equals 735 and work our way up. So working mod 539873 we find 735 squared congruent to 352, which factors nicely as. And maybe we'll get lucky and find our necessary factors immediately. Don't count on it. So we find 736 squared and that gives us, and I'm not sure what the factors of this are, but there aren't any factors less than or equal to 19, so we'll go on to 737 squared. And we can start our factorization, but since a prime factor of 103 is greater than 19, we don't really need to finish the factorization here. We can just go on to the next number. So we'll continue trying larger and larger numbers. And after some work we find 750 squared, that's 22627 is. After some more work we find that 783 squared is 801 squared is 101728, which factors as. And we keep finding these numbers until, what's that? A portal of the spacetime continuum has opened up. Hello, earlier self. This is your later self saying that you only need to find these four numbers. Oh, that's good to know. Thank you, later self. So if your later self has a way of communicating with your earlier self, well, he's kind of a jerk for not giving you stock tips. But in any case, you'll know when to stop because your later self will tell you. Now, if you don't have access to a handy time machine, we keep going until we decide not to, and if it works great, otherwise we'll have to keep going. Now it's important to recognize that none of these numbers are perfect squares. So what we want is we want a product of these numbers to give us a perfect square. Since we know the prime factorization of 352, 22627 and so on, we can write this in prime factored form, multiply them all together. And now for the product to be a perfect square, all of our exponents must be even. So we're going to require working mod 2. The exponent at 2 has to be congruent to 0. The exponent of 11 must be congruent to 0. And similarly for the exponents on 13 and 17. And so now we look for a non-trivial solution. And so that means we need this first factor, 352, and the fourth factor, 101728. So remember, 352 is 735 squared, and 101728 is 801 squared. Since we have the prime factorizations of 735 squared and 801 squared, we can write the product in prime factored form. Now if I rearrange this equation, we find that 735 times 801 squared minus 2251117 squared is going to be congruent to 0 mod 539873. And since this is a difference of perfect squares, I can factor the left-hand side, and that tells me the greatest common divisor of 735 times 801 plus or minus 225 times 11 times 17, and 539873 will actually be a factor of our modulus. And since the difference will generally give us a smaller number, we'll find the greatest common divisor, which is 1949, and since that's one factor, we can find the other by direct division, giving us a factorization. Finally, to avoid creating a paradox that will accidentally destroy the space-time continuum, we have to go back in time and tell our earlier self that we could have stopped with these four numbers. So just a moment, I'm going to open up a portal in the space-time continuum. What's that? A portal in the space-time continuum has opened up. Hello, earlier self. This is your later self saying that you only need to find these four numbers. Oh, that's good to know. Thank you, later self. So again, with a handy time machine, we can know that we only need these four numbers to find the factorization. Of course, if you don't have a time machine, then we'll have to keep going until we find a factorization.