Last night, as I was in bed trying to get to sleep, I thought about the factor field thing I’ve been messing around with. As a result, it took a couple hours to fall asleep :-) But in the process I realized something.
Up until now, I’d been saying that using the factor field you can see which numbers are prime numbers up to row 256. This is because the width of an Excel spreadsheet can only be 256 max. However, as I was trying to drift off to sleep, I realized that we can know a lot more than this!
If a number is a factor but it shows up off the edge of the Excel sheet, we know that there is a smaller factor that will show up. For instance, 512 has the number 256 as a factor; but it also has the number 2 as a factor (as 2 x 256 = 512). As a result, the existence of the smaller factors will let us know whether a factor larger than 256 exists. And this holds all the way up to the square of 256. In other words, anything with factors smaller than 256 will show up on the factor field–or to put it another way, the smallest factor of a number that will not show up on the factor field must be greater than 256.
This means that we can determine prime numbers up to a number that contains the factor 256 x 256 (the smallest possible factor that would not show up). 256 x 256 = 65536, which happens to be the highest row in the Excel spreadsheet. In other words, any line on the spreadsheet that does not have more than 1 factor (after line 256) is prime.
Tonight I will try not to think about any of this and see if I can actually sleep :-)
