So, I've been working on this for a while trying to attack it from different angles. I've identified a couple of rules governing the factoring of numbers, Im not sure if this is new stuff or not… For every number (except for the square of every second even number) has a perfect square number larger than it that when subtracted from that larger perfect square number results in a smaller perfect square. The root of that larger perfect square is always the average of the sum of the factors of the starting number. The root of the smaller perfect square is always the distance between the average of the factors and the factors themselves.

For instance:

7 *11 = 77

7+11=18

18/2=9

9*9=81

81-77= 4

9-2=7

9+2=11

This works with all odd numbers and every other even number. And it also always seems to work with any number that is the product of two primes.

Unfortunately, its just as hard to predict the average of the two factors as it is to predict the factors themselves. But, I noticed something else:

For almost every number there is only 1 larger number that is 1) a perfect square and 2) the difference between the two is also a perfect square. So, I've been thinking that since every subsequent square number is the root of the next lower square * 2 +1… I think it should be possible to plot an intercept course.

Sort of like the old 'police pursuit' physics problem but with the diffence being, the two cars travel a constant speed, and the distance "between stops" is what changes. Going off of the example of 77 above, car 1 starts at 77, car 2 starts at 0. If they travel the same speed and only stop at square numbers then the next time they stop at the same time is when car 1 is at 81 and car 2 is at 4.

TLDR; is there a formula that could be used to plot an intercept course where speed is constant, the 2 objects in motion make stops where the distance between stops increases by a known variable and the point of interception is when they both stop at the same time?