Still working on the d[t] and a[t] analysis. I have been looking at quite a bit of data, trying to make heads or tails of this. Would appreciate anyone taking a look and letting me know if I have gone too deep down the rabbit hole. (or not deep enough.)

Latest version of d[t] and a[t] analysis with a new "summary" at the top is attached. This is again for c=145.

The negative t records have been removed to save space. The results I am showing are mirrored in negative t.

The summary at the top is an attempt to condense relevant information and make it easier to review multiple test cases.

Which will follow in separate posts.

The summary columns are as follows:

(d[t]-d)/(n-1): values where n-1 is a factor.

(d[t] diffs): difference between values.

(x diffs): difference between x values where n-1 is a factor.

(t diffs): difference between t values where n-1 is a factor.

(d[t] diff)/t: Only shows where t is a factor of d[t] diff. (An attempt to find a common basis.)

(a[t]/n): values where n is a factor.

(a[t] diffs): difference between values.

(x diffs): difference between x values where n is a factor.

(t diffs): difference between t values where n is a factor.

(a[t] diff)/t: Only shows where t is a factor of a[t] diff. (An attempt to find a common basis.)

(1/4)

>>3456

Pics attached are d[t] and a[t] summaries for different values of e, odd and even n.

c=441

an example at e=0 for a=3^2, b=7^2.

c=143 and c=145

examples at even and odd e where n is odd.

c=115 and c=481

examples at even and odd e where n is even.

(2/4)

>>3457

Pics attached are side by side comparisons of the d[t] and a[t] analysis for various values of c and their prime solutions.

Samples are for c=145, 533, 785, and 901.

Results are run over the same list of records. All starting from (e,1,1) and up to t=c.t + c.n.

(3/4)

>>3458

If you've made it this far, following are my current thoughts:

1) We can accurately identify 3 records in each of the (d[t]-d) and (a[t]) sequences. This applies only to the initial 1,c record. Don't know the value of this yet, but it is there.

For (d[t]-d): 1, xx-f, and c.

For a[t]: 1, xx-f+4, and c.

2) For the initial 1,c records, some records have t as valid factors of the d[t] diff or the a[t] diff. When these factors exist, the pattern appears to be:

For even n:

d[t] diff / t are even (2,4,6,etc)

a[t] diff / t are sequential (1,2,3,4,etc)

For odd n:

d[t] diff / t are sequential (1,2,3,4,etc)

a[t] diff / t are even (2,4,6,etc)

Don't know if this is important or not.

3) One of the sequences of values in a[t]/n uses x.

4) One of the sequences of values in (d[t]-d)/(n-1) uses (x+1).

Still have no clue what offset we are looking for.

(4/4)

>>3675

Teach. Looking into this now also.

Sample for c=481 attached. I added (x+n) just as conceptual for which nodes get included.

>>3677

Also added prime factors for d[t] and a[t] analysis. See pic attached for c=481.

The d[t]-d primes column really clarified for me where the values are we are searching for.

Also, if you look at the (40,1,209) record, the d[t]-d value can be calculated by (xx-f)*(n-1). Perhaps the tree factors are a way to simply remove certain factors from the value?

>>3692

Try just removing trailing zeros from the d values.

>>3730

don't think so. prime solution for 481, n = 4. (40,4,5) = {40:4:21:8:13:37} = 481

>>3830

Pic attached is a tree example of c=921 with and without trimming values.

In the top example, 30*30 + 21 == 921 as expected.

The bottom example, the d node is created trimmed in a recursive call, and that trimmed value is used for the d and e assignment and so on.

Not sure if we have an answer if the nodes should always be in balance with the formula.

In other words, should be trim before or after the sqrt(d) ?

>>3869

>>3887

Baker - thanks for these additional clues. Still plugging away.

I've taken your idea of triangles and inverse triangles and incorporated them into a parse tree. Tried to find combinations that will get us back to any solution.

Pic attached is just latest attempts adding all d's, e's, and just the d triangles that are direct descendants from c.

Working in (0,1) space for c=145, I was able to match an x value of 120 by 4(24 + 6). Also managed to find the factor of 5 by adding triangles sqrt( 2(1+6+78) - 145 ).

Of course, this doesn't work in too many places.

Interesting that things keep coming back to difference of two squares.

For example: 278 - 26 = 144 + 1 = 145.

If we are searching for x or x+n (x+n is just x+1 in (0,1) space), then we're looking for the correct level in the tree, not a factor.

Does this parse tree tell us what level we are at or the level we are looking for? Does the triangle value or difference in triangles between branches of the parse tree tell us this?

>>3896

I wrote a test a little while ago to see if there was any pattern to prime numbers in the grid. Obviously haven't figured that one out, but it certainly looks possible.

>>3924

The only important thing to note with those code lines is the recursive call with the prefix string. That's what gives the indentation in the output.

>>3909

Thanks for the very clear example, Teach. I've updated my tree parsing to incorporated the /2 step for clarity.

VQC has hinted that the tree will lead to an x or x+n solution. Still searching in the dark for this, but ran some sample code against a list of prime numbers.

Pic attached is for prime numbers 509, 521, and 523.

If you isolate the d branch in the tree and calculate the triangle values for each d node, the total is really close to the (x+n) value.

Well, close enough to be interesting.

These examples share the same d branches and the e branches vary slightly, perhaps there is something to learn here.

>>4047

VA. I added f into all the d nodes in the tree, and am reviewing against (e,1) entries to see if anything obvious jumps out.

This has to relate back to the d[t]-d hint at some point. But not seeing anything.

>>4049

Still believe in the simple and elegant. Trees are pretty elegant, after all.

Unfortunately, still walking around in the dark and hitting walls at every turn. Guess that's what makes it fun. Learning something that hasn't been taught before.

Would love to hear the back story to this one day. It must be fascinating.

>>4065

Ready!

>>4066

Very interesting. Thank you.

>>4080

Running out of ideas with the tree.

Went back to the grid and added prime factor output to various test cases for d[t]-d and a[t] values. Just further researching triangle numbers. c145 example attached.

Also, the hint "if a number at position t has a factor s at (e+1)" makes more sense when you break down the values into their prime factors. Sample for e=2, d, and the d prime factors also attached.