Now we look at values of x from 783 to 802 (these are values centered about 793.27..., the square root of 629287). We mark which primes divide Q(x) for each value of x

Now we try to factor Q(x) = x² - 629287 for each value of x, using the table to tell us which primes will factor Q(x). Since most of the entries in the table are blank (81.5% to be precise), we save a lot of work by using the "sieving" methods of building the table to tell us which primes we need to consider. For larger numbers, the savings are even greater (much greater). Finally, after trying the twenty different values of x in the table, we find three 29-smooth values of Q(x).

Next we look at the polynomial Q₂(x) = x² - 2´629287. We repeat the process of finding which primes from 2 to 29 might divide Q₂(x) and build a table for values of x from 1111 to 1130 (this range being centered about the square root of 2´629287). Then we go through the process of trying to looking for values of x which give us 29-smooth values for Q₂(x). We find two more

Today (Friday) you are to work together to see how many more 29-smooth values you can find as a class. I will be back on Monday and I’m expecting to see several more values on your list. Good luck.