All quiet in here, so I'll post my findings for today. See if you guys can take this a bit further.
Started with a quest to see if I could walk the tree to various n records by substituting a and x from one to the next.
I iterated a number of values of t in (1,1,t) and jumped to the next n level.
Unfortunately, that led me in a circle where, for example:
From (1,1,2) = {1:1:8:3:5:13} = 65; (x+n)^2=4x4=16; (d+n)^2=9x9=81.
jumping by n=a:5, x=x:3.
to (1,5,2) = {1:5:4:3:1:17} = 17; (x+n)^2=8x8=64; (d+n)^2=9x9=81.
then the next jump by n=a and x=x sent me right back to (1,1,2)
However, in doing those tests, I noticed a small square pattern that might be of some use.
Following is a test I ran for c = 785, with a=5 and and b=157.
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TestPrimeResult for a=5, b=157.
From (1,365,14) = {1:365:28:27:1:785} = 785; (x+n)^2=392x392=153664; (d+n)^2=393x393=154449.
jumping by n=a:1, x=x:27.
to (1,1,14) = {1:1:392:27:365:421} = 153665; (x+n)^2=28x28=784; (d+n)^2=393x393=154449.
difference between to.c=153665 and from.(x+n)^2=153664 = 1.
to.c - from.(x+n)^2 == e? 1 - 1 = 0.
From (1,53,12) = {1:53:28:23:5:157} = 785; (x+n)^2=76x76=5776; (d+n)^2=81x81=6561.
jumping by n=a:5, x=x:23.
to (1,5,12) = {1:5:76:23:53:109} = 5777; (x+n)^2=28x28=784; (d+n)^2=81x81=6561.
difference between to.c=5777 and from.(x+n)^2=5776 = 1.
to.c - from.(x+n)^2 == e? 1 - 1 = 0.
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Explanation as follows:
From: (1,365,14) is the a=1 entry point for c = 785.
To: (1,1,14) is the record created by variables e, n, x where n=(1,365,14).a and x=(1,365,14).x.
From: (1,53,12) is the prime solution record we are looking for where a=5 and b=157.
To: (1,5,12) is the record created by variables e,n,x where n=(1,53,12).a and x=(1,53,12).x.
Patterns:
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There is a consistent relationship between the From record and the To record that can be represented by the formula:
to.c - from.(x+n)^2 == e;
The difference between the destination c value and the source small square is equal to e.
This appears to be true for all "jump" records where a to.n = from.a;
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The size of the small square in the jump record from a=1 entry point is equal to the small square size of the jump record from the prime result record.
(1,1,14) small square = 784.
(1,5,12) small square = 784.
What's the big deal?
I think there may be a solution in here somewhere. Using the following steps.
1) P(e,n) at a=1 for any c.
2) Pj(e,n) where n=P(e,n).a and x=P(e,n).x.
3) use the small square equality to figure out the pj(e,n) record. We now know e and the value of (x+n)^2. Perhaps some fancy algebra can give us values for x, n, or d.
4) reverse jump the pj(e,n) record to p(e,n). We know e, d, and something else from step 3.
If this makes any sense to anyone that would be awesome!!!