Quadratic Sieve ...

Every mathematics student likes primes numbers. Every mathematics student likes parabolas. Let's put them together to have some fun:
Consider two natural numbers a and b. Of course (-a,a^2) is on the graph of y=x^2, as is (a,a^2). Give two points to a math major and the odds are that he or she will connect them with a line. Give a line to a math major and the odds are he or she will compute the slope (not useful to us here) and the y-intercept (what we really need). What is the y-intercept of the line connecting these points? Use this to construct another prime number sieve.