I think we should be looking for a D value. Specifically the one where n = 0. Also where -x^2 = e Then, if we have that D value we can look at (d+n)^2-(x+n)^2 = c to get just d^2-x^2 = c. The thing is that for our first factors (1,c) we will not have n=0 here it will be something else. Another thing I noticed is that if you make a record (e,n,d,x,a,b) then make a new record with the same a,b but make the new d value equal to the old n value, then your new n value will be the same as your old d value, so in a way you can switch d and n but the x and the e will change. Then another thing is that VQC mentioned the record c^2. This is cool because if you shift it to n=0, then you can get d^2 - x^2 = c^2, a pythagorean triple (learned in number theory today you can generate these from a different triple. Could maybe just swap c and x in this). This would also mean that (d+x)(d-x) = c^2. This could only be true if x = 0 in which case d=c. Or this could make the scenario where (assuming b>a)

(d+x) = ab^2 and (d-x) = a

(d+x) = ba^2 and (d-x) = b

(d+x) = b^2 and (d-x) = a^2 [I like this one best]

then

d + x = b^2

d - x = a^2

2d = a^2 + b^2

We know the correct D for our original record is the one where D = (b+a)/2. 2D = b+a

Also for this the correct X would be equal to (b-a)/2.

(b+a)(b+a) = 4D^2

b^2 + 2c + a^2 = 4D^2

2c + 2d = 4D^2 =c + d = 2D^2

b^2 + a^2 = 4D^2 - 2c

b^2 + a^2 = 2D^2 + 2(D^2 - c)

b^2 + a^2 = 2D^2 - 2E = 2D^2 + X^2 = 2(D^2 +X^2)

b^2 + a^2 = 2(D^2 + X^2) = 2d

d = D^2 + X^2

from this we can infer that the correct d value for the record where (d-x)=a^2 and (d+x)=b^2 is also a square. Lets see if we can figure it out from here:

( * is for square records, capital letters are the goal record.)

d* = D^2 + X^2

a* = a^2 = (D-X)^2 = D^2 - 2DX - X^2

x = d - a = D^2 + X^2 - (D^2 - 2DX - X^2)

= 2X^2 + 2DX

b = b^2 = (a+2x+2n)^2 =(D^2 - 2DX - X^2 + 2(2X^2 + 2DX))^2

= ( D^2 - 2DX - X^2 + 4X^2 + 4DX )^2 = (D^2 + 2DX + 3X^2)^2

c* = c^2

n* = 0

e = c - d*^2 = c^2 - (D^2 + X^2)^2

= (D^2 + E)^2 - (D^2 + X^2)^2 = (D^2 - X^2)^2 - (D^2 + X^2)^2

= D^4 - 2D^2X^2 + X^4 - D^4 - 2D^2X^2 - X^4

= - 4D^2X^2