Damn my intuition bells are going off right now Lads! What I’ve found in using my method is that when you begin building up the factors of the (x+n)^2 -1 area, you first arrive at a perfect square that is a potential match for Potential x+n or P(x+n). D, e, and f all determine what the first perfect square will be, and then

The big idea is finding the first perfect square built using the root of f.

And then using multiples of it.

F limits the possible correct values.

It’s a 2 step process. All larger x+n values can be built from the first perfect square.

Once you find the first perfect square, the other possible values are just multiples.

So PTB for x+n=6889 = 2(sqrt ((f-1)/8))-1 = 7

First match is 7 * 5 +1 = 36

Sqrt(36) = 6 = potential x+n

Here's the perfect square creation I'm talking about. This is building the x+n square using 4 rectangles (or 8 triangles) +1

Step 1: find the first perfect square.

PTB = 7

7 * 5 +1 = 36

potential triangle base value(u) is 6

potential triangle side value (u-1) is (6-1)=5

Step 2: correct x+n is a multiple of the correct value above ^^^

Formula is: u * (u-1) * 4 +1 = potential x+n

1. 6 * 5 *4 +1 = 121, sqrt(121)=11

2. 12 * 11 * 4 +1 = 529, sqrt(529)=23

3. 18 * 17 * 4 +1 = 1225, sqrt(1225)=35

4. 24 * 23 * 4 +1=2209, sqrt(2209)=47

5. 30 * 29 * 4 +1= 3481, sqrt(3481)=59

6. 36 * 35 * 4 +1= 5041, sqrt(5041)=71

7. 42 * 41 * 4 +1= 6889, sqrt(6889)=83 = correct x+n value

Step 3: use the quadratic to check/verify each iteration above ^^^ resulting in an integer answer (no remainder)

Sqrt(c + (x+n))-d = n

I'll run it on some more c values, I know this is c6107.

Just trying to find an idea that makes sense first, and then test it out.

Pretty cool to do this from memory with a calculator and pad. I love this problem. I think about it in the morning when I wake up, and it pops into my mind during the work day. I haven't posted much lately bc I had nothing new, but maybe this idea is worth exploring.