>>2574

Was enjoying the -x discussion earlier and tinkering in the background.

I have made a little bit of progress, but nothing that I'd call finished.

Pictures attached are of the (0,1) and (1,1) -x space for c=65.

In the left columns, you'll note (abc), (a), (b). These represent where the a, b, or c values from our prime solution are factors of the a value in the record.

Interestingly, if you navigate directly to (e,n,t) at (0,1,-c), this is the first occurrence of an a value that has abc as factors.

Based on these pictures, I don't think we have to do much searching at all. The a factors are literally everywhere. Pulling them out is a different story.

>>2604

>>2606

a(e,1,1) = c

can be found at both:

(2c,1,1)

(2c-1,1,1)

>>2603

(e,1,1) = c I'm having a bit more trouble finding.

>>2608

shit… any here I was thinking all my code was buggy.

So you're confirming that the relevant records are at (2c,1,1) and (2c-1,1,1) correct?

Do you have any additional insight into how they are useful?

>>2612

my understanding: a(e,1,1) means the a value for the record (e,n,t)

>>2614

dude… wasn't me. But I like your notation better!

quick question. if we have multiple records with a values that share factors a, b and c, all with different values.

Then divide the c out, we're left with some combinations of a*b.

With 2 records, with different remainders, can we solve for a?

>>2619

>>2620

ok. Just an example:

for c=145

a of (290,1,-289) = 168345

a of (289,1,-290) = 168925

both share a,b and c as factors.

can we discern anything from this?

There are so many ways of getting to records that share a,b,c as factors.

>>2622

I understand, I expanded the search to 10cn just to pick up more data.

>>2623

yeah, screen shot was testing cc, 2c, 2c-1, and various other (e,1) combinations.

>>2628

thanks. back to digging.

is it possible that the records we are searching for in (e,1) have a values equal to either a^2 or b^2?

>>2636

scratch that. doesn't apply everywhere.

>>2638

>>2639

ok. here's what I'm looking at, still for c=145.

starting position at (e,1,1) where a = n*c

(1,1,67) = {1:1:8978:133:8845:9113}

factors of a are a,b,c

negative x move from original (1,61,4)

(1,1,-66) = {1:1:8712:-133:8845:8581}

factors of a are also a,b,c

These search boundaries can be created for any (e,1).

Between these t=67 and t=-66, there are 2 instances where a = a^2 (at (1,1,-3) and (1,1,4)), and a = b^2 (at (1,1,-20) and (1,1,21)).

Not sure what to make of these, as I didn't find them in other examples.

>>2653

>>2654

playing along. I'm with you on the up step. not so much on moving down. Are you manipulating something other than x? If so, can you share that formula?

also, your 290 example isn't a valid record.

c != ( d + n ) * ( d + n ) - ( x + n ) * ( x + n )

21025 - 20736 = 289.

>>2657

Thanks.

fyi, an alternate to moving up:

BigInteger newA = ter.b;

BigInteger newD = 2 * (ter.b - ter.d) + ter.d;

TheEndRecord up = TerFactory.CreateForNDA( ter.n, newD, newA );

I got similar exceptions with other even values.

And what logic are you using to resolve a and b?

>>2658

I think it might be.

>>2661

I thought you had taken it a bit further.

I've playing a little in the (e,1) space trying to find more patterns and/or relevant records.

pic related show output for c=65, 145, and 785 and only includes records that meet the criteria:

1) a = na, nb, or n*c from the c entry record (indicated as na, nb, nc); or

2) a = na, nb, or n*c from the p solution record. (indicated as pa, pb)

as we know, the t values for these records are n apart. And we can see that extend into negative t.

I've also indicated the relationship between a and the prime a, prime b, and c values.

What's interesting is that none of the "prime" records are divisible by c. So unless the search scope needs to widen beyond n*c, we can't rely on c to find our answer.

Was looking into some relationship between f and c+f to see if there was another way to navigate here. No such luck yet.

>>2735

VA - still trying…

I've put a lot of output together, still trying to find any pattern that links to any potential prime match. No such luck yet. Notwithstanding the -x hint earlier. All I've been able to confirm is that the pattern holds for na, nb, nc in negative x and f spaces.

I had a thought earlier that I've been trying to work through in the (e+1) space. Perhaps someone can assist or tell me if I'm barking up another wrong tree…

VQC mentioned that every factor for c exists in (e+1). I originally thought that meant (e+1,1), but I'm leaning now towards (e+1,n). And we also know that the a value at (e,n) moves to the d values at (e+1,n). (I believe this is correct).

For example:

for c=145 (1,61,6), the factor 5 exists as the d value of (2,1,2).

for c=901 (1,421,15), a factor of 17 exists as the d value of (2,33,8).

I thought that if there was some way to link records to e+1, we would have a simpler way to calculate d?

new avenue to explore, perhaps.

Records at c and ab both share f and c+f values.

They also share a common record at (e,1, (c+1)/2) which has a different f value.

If you move the 3 records to (e+1, n, t) they each now have different f values. Is there a way to find a difference between these 3 records?

>>2757

interesting. thanks.

(1,61,6) = {1:61:12:11:1:145} = 145

(1,61,56) = {1:61:212:111:101:445}

(1,61,67) = {1:61:278:133:145:533}

The following formula works for generating these next records for (e,n).

a=a+2*(x-n) will generate (1,61,56) - this one can also come from (e,n,t+n-x)

a=a+2*(x+n) will generate (1,61,67) - this one can also come from (e,n,t+n)

My previous move by d formulas stepped over (1,61,67).

>>2762

not sure if this is of any value, but I found a few more records for (1,61) at -t.

(1,61,-66) = {1:61:12:-133:145:1} = 145; f=24; (x+n)=-72; f+c=169; -x: 11

(1,61,-55) = {1:61:-10:-111:101:1} = 101; f=-20; (x+n)=-50; f+c=81; -x: -11

(1,61,-5) = {1:61:-10:-11:1:101} = 101; f=-20; (x+n)=50; f+c=81; -x: -111

(1,61,6) = {1:61:12:11:1:145} = 145; f=24; (x+n)=72; f+c=169; -x: -133

(1,61,56) = {1:61:212:111:101:445} = 44945; f=424; (x+n)=172; f+c=45369; -x: -233

(1,61,67) = {1:61:278:133:145:533} = 77285; f=556; (x+n)=194; f+c=77841; -x: -255

>>2764

if you are referring to the rowNegX method, yes, I'm familiar with what you're doing. I have some sample code that follows along.

I'm not quite understanding what you mean with p. How is it different to the x or t values that we already know, or how we can navigate between records by factors?

>>2773

back to understanding (n-1)*a movements.

Pic attached is of c=156. Record starts at (1,61,6), I've moved the analysis to t + n at (1,61,67), and included the factor records at (1,29,67), (1,5,67) and differences between each and differences to (1,1,67).

As expected, they all follow a pattern. d and a move together. The f difference is always 2d difference. And (n-1)a applies when moving to (1,1).

There is also a pattern to moving from (1,61, 67) to factor records.

>>2775

Oops. C=145

>>2784

yes.

I also thought that the d difference between (1,29,67) -(1,5,67) equals (1,61,6) n * 2d or n*f. But that formula doesn't apply to different test cases.

So for these records, if we can figure out the a or d formula…

>>2786

I've also looked into a formula for b. Haven't found a combination of x+n, x-n, multiple of n, etc. that will work.

Interesting thing about these records is the shared e and x. and the 2an is all the same - as you would expect.

Pity we don't yet understand moving between n's other than to 1.

So the (n-1)*a takes you to the n=1 value and is a shortcut for the "f transition" (whatever that means).

What's the formula to take you to another n value other than 1?!???

Guess that's the Integer Factorization secret…

>>2786

important formula I believe:

n = (xx + e) / (f - 2x)

>>2791

my equation worked for c=145. I'm trying to get a formula for n in terms of known variables. e, x, d movement we have some formulas for already.

>>2791

>>2793

works for c=65 also.

more testing required for e other than 1.

n = (xx + e) / (f - 2x)

n = (xx + e) / (2d + 1 - e - 2x)

if this holds, we have a formula for n in terms of known x and e variables. And now we need to understand better the d jump.

>>2798

>PMA, remember "Use the z on y at x" ??

VA - I still have no idea what this means…

>>2802

I've been trying to get further along with the (n-1)*a hint.

I'm starting with a record for c at (e,n,t). I move up to a record at (e,n,t+n). This record contains all factors of a,b,c.

It is also similar to a record at (e,1,t+n) in that e and x are the same.

Then if you find records at (e,a t+n) and (e,b,t+n) where a and b are the values we are searching for, you'll notice that they also share e and x values. And 2an are the same value for all of these records.

We know how to take our (e,n,t) for any c and translate it to (e,1,t). That's the (n-1)*a transformation.

So I've been researching relationships between these records to try and learn how to transform the n value to a or b instead of 1.

In analyzing those differences, and searching for a formula to define the change in a or d, I stumbled upon a way to represent that change in terms of known variables for e, x, and d.

Those are the formulas posted at >>2794.

Unfortunately, they only appear to work where e=1.

>>2804

no. it gives you any n from any (e,1) and it may provide a way to get to n for a,b in terms of e, x and a change in d values.

>>2807

I may not be explaining myself. That formula doesn't solve for n yet.

It is a step I am using to break down variables into smaller pieces to try and analyze the jump to a valid d that will solve for n.

>>2809

sure. I'll look into it.

Continue some research into records calculated from c into (e,n,t+n).

These are labeled nt in the attached pic.

Can someone please tell me why (b-1)/x is always equal to 4?

>>2812

And if you compare the b values between t=n+t and t=(1-(n+t)) records, the difference also appears to be 4x.

Example

(1,5,67).b - (1,5,-66).b = 2045 -1513 = 532 = 4x133;

(1,29,67).b - (1,29,-66).b = 629 - 97 = 532 = 4x133;

>>2821

yes. for moving right or left.

>>2822

thanks.

>>2824

Look at the attached image in >>2815

The mirrored (e,n) at x and -x follow a pattern where the difference in x is always 2x, and the difference in b is always 4x.

The equations make this obvious. Just restating that there are 2 possible ways to get to a solution, and 2 hints about movement in b.

>>2827

>"Use the z on y at x"

this hint was from VQC, correct?

>>2831

thanks. I've reread. Still searching…

>>2834

>"Use the Y on C at X."

this is the correct quote. it also mentioned the boiling point of vinegar. 100.6c or 213f. Mentioned 1,1,2,3. Some other stuff I didn't understand.

and this gem also:

112115 is also 5510101020

which I think means {1:1:2:1:1:5} is also {5:5:10:10:10:20}?

>>2835

>{5:5:10:10:10:20}?

doesn't look like a valid record. no clue.

Guys… A little bit of progress.

pic related. See "connection?" record.

Only works in 3 of my test cases. Perhaps we can take it further.

Process:

1) create c record.

2) create record at (e,1,1-(n+t)). Only need this for the negative x value.

3) create new (e,n) record with EDA where:

newd = (n-d)3 + 2d + 1

newa = newd - xFromStep2.

4) the n value in the record from step 3 is the b value in the prime result. Solve using EDB.

>>2840

t comes from the c record at (e,n,t)

>>2838

>>2842

Create the c record normally. Using the values from there to create another record via ent:

e = c.e

n = c.n

t = (1-(c.n+c.t))

This will give you a -x value.

Alternatively, create a record via ent with t equal to n+t, and just use the x value as negative.

>EDB, EDA

just some helper functions to create records.

public static TheEndRecord CreateForEDA( BigInteger e, BigInteger d, BigInteger a ) {

BigInteger c = d * d + e;

BigInteger b = c / a;

BigInteger n = ( a + b ) / 2 - d;

BigInteger x = d - a;

return new TheEndRecord( e, n, d, x, a, b );

}

public static TheEndRecord CreateForEDB( BigInteger e, BigInteger d, BigInteger b ) {

BigInteger c = d * d + e;

BigInteger a = c / b;

BigInteger n = ( a + b ) / 2 - d;

BigInteger x = d - a;

return new TheEndRecord( e, n, d, x, a, b );

}

>>2844

you're welcome!

By the way, we can save a calculation at the beginning (if this ends up being a correct approach), by creating our entry record with a,b values of c,1.

This gives us the same e, n, d values, but with the -x we are looking for.

example:

145=5x29

created from a=1, b=c

c = (1,61,6) = {1:61:12:11:1:145} = 145;

created from a=c, b=1

-c = (1,61,-66) = {1:61:12:-133:145:1} = 145;

>>2848

There are similar entries in the -n space:

(-4,0,2) = {-4:0:4:2:2:6} = 12

(-9,0,2) = {-9:0:5:3:2:8} = 16

I don't see one for 2, 7 though.

>>2849

>There are similar entries in the -n space:

-e space.

The formula to go to matching -x records depends on e:

even values of e

(e,n,t) -(e,n, (2-t))

odd values of e

(e,n,t) -(e,n, (1-t))

Still working. Rereading crumbs.

>A prime number appears once in a column. This is SIMPLE to calculate. This calculation is important. I call n for a prime number big_n or N

>The product of two primes appears twice. Once the same as the prime, which is easy to calculate and the second time, this is the n that we are after.

The simple to calculate comment was with regards to the entry point at a=1,b=c.

Anyone made any further progress?

>>2858

>Root of d = {0, 2d, 3d, 2d, d, 9d}. All a and b of ( 0, 2*d, t) are multiple of d for all t.

>Yes and that pattern can be used elsewhere.

See pic related of c=145 where d=12 (both c and prime result records), for (0,2a), (0, 2b), (0,2n), (0,2c)

p e=0 n=2a a=a =(0,10,6) = {0:10:15:10:5:45} = 225;

p/c e=0 n=2d a=d =(0,24,13) = {0:24:36:24:12:108} = 1296;

p e=0 n=2b a=b =(0,58,30) = {0:58:87:58:29:261} = 7569;

c e=0 n=2n a=n =(0,122,62) = {0:122:183:122:61:549} = 33489;

c e=0 n=2c a=c =(0,290,146) = {0:290:435:290:145:1305} = 189225;

>>2870

and c for these records is d^2.

>>2872

Here are valid (0,10,t) records from t=-10 to t=50.

They only exist 5 t apart.

(0,10,-9) = {0:10:0:-20:20:0} = 0; f=1; (x+n)=-10; f+c=1; -x: 0; (x-n)=-30

(0,10,-4) = {0:10:-5:-10:5:5} = 25; f=-9; (x+n)=0; f+c=16; -x: -10; (x-n)=-20

(0,10,1) = {0:10:0:0:0:20} = 0; f=1; (x+n)=10; f+c=1; -x: -20; (x-n)=-10

(0,10,6) = {0:10:15:10:5:45} = 225; f=31; (x+n)=20; f+c=256; -x: -30; (x-n)=0

(0,10,11) = {0:10:40:20:20:80} = 1600; f=81; (x+n)=30; f+c=1681; -x: -40; (x-n)=10

(0,10,16) = {0:10:75:30:45:125} = 5625; f=151; (x+n)=40; f+c=5776; -x: -50; (x-n)=20

(0,10,21) = {0:10:120:40:80:180} = 14400; f=241; (x+n)=50; f+c=14641; -x: -60; (x-n)=30

(0,10,26) = {0:10:175:50:125:245} = 30625; f=351; (x+n)=60; f+c=30976; -x: -70; (x-n)=40

(0,10,31) = {0:10:240:60:180:320} = 57600; f=481; (x+n)=70; f+c=58081; -x: -80; (x-n)=50

(0,10,36) = {0:10:315:70:245:405} = 99225; f=631; (x+n)=80; f+c=99856; -x: -90; (x-n)=60

(0,10,41) = {0:10:400:80:320:500} = 160000; f=801; (x+n)=90; f+c=160801; -x: -100; (x-n)=70

(0,10,46) = {0:10:495:90:405:605} = 245025; f=991; (x+n)=100; f+c=246016; -x: -110; (x-n)=80

Maybe this means that (0,2d,t) records only exist where t = d apart.

>>2874

not quite correct. Examples where d = 12 at (0,24). Valid records are 6 apart.

(0,24,-23) = {0:24:0:-48:48:0} = 0; f=1; (x+n)=-24; f+c=1; -x: 0; (x-n)=-72

(0,24,-17) = {0:24:-9:-36:27:3} = 81; f=-17; (x+n)=-12; f+c=64; -x: -12; (x-n)=-60

(0,24,-11) = {0:24:-12:-24:12:12} = 144; f=-23; (x+n)=0; f+c=121; -x: -24; (x-n)=-48

(0,24,-5) = {0:24:-9:-12:3:27} = 81; f=-17; (x+n)=12; f+c=64; -x: -36; (x-n)=-36

(0,24,1) = {0:24:0:0:0:48} = 0; f=1; (x+n)=24; f+c=1; -x: -48; (x-n)=-24

(0,24,7) = {0:24:15:12:3:75} = 225; f=31; (x+n)=36; f+c=256; -x: -60; (x-n)=-12

(0,24,13) = {0:24:36:24:12:108} = 1296; f=73; (x+n)=48; f+c=1369; -x: -72; (x-n)=0

(0,24,19) = {0:24:63:36:27:147} = 3969; f=127; (x+n)=60; f+c=4096; -x: -84; (x-n)=12

(0,24,25) = {0:24:96:48:48:192} = 9216; f=193; (x+n)=72; f+c=9409; -x: -96; (x-n)=24

(

>>2878

>>2877

I have no doubt the solution is in plain sight. And something very simple and elegant.

But those are not valid t values for d=5, unless t needs to be redefined based on column (0,n). Which I think is an incorrect approach.

>>2878

Starting looking at (0,2d,c) and (0,2d,n) for c=65 and c=145.

Interesting that they both exist.

>>2871

d = {0, 2d, 3d, 2d, d, 9d}

I think the c for these records is 9 * d^2.

The c value at the record (0,2d) where a = d, is equal to 9*(c-1).

Example for c=145:

(0,2d) record:

(0,24,13) = {0:24:36:24:12:108} = 1296

(e,n) for c=145 record

(1,61,6) = {1:61:12:11:1:145} = 145

9*(145-1)=1296

I think generally, this can be written as:

c of (0,2d) at a = d

equals

9*(c-e) at (e,n) at a=1, b=c

I have looked at various records at e=1, 5, 6 and 8. Someone please confirm.

As the prime results share a relationship to 2d, perhaps there is a connection to those records as well?

>>2962

en.wikipedia.org/wiki/Thue–Morse_sequence

T3 = 01101001.

T4 = 0110100110010110.

0=R, 1=L

for any music anons, looks like a paradiddle.

>>2936

duh. ab=c; a=d, b=9d, c=9(d^2).

continuing with (0,2d) analysis. Think I've found the boiling point of vinegar reference.

pic related shows movement of (0,2d) record to (0,1) using (n-1)*a transform, then to (1,1) using boiling point of vinegar transform.

Very funny. Not sure how useful in solving our prime problem.

Theory:

a) at (e,n) there exists a record where (b-a) = 2*d.

b) the n for that entry represents the a value in the prime solution we are looking for.

c) the d in the (b-a) = 2*d formula comes from the d in our initial c record.

Pic attached shows example at c=785.

In plain sight, perhaps in the c=145 example solution record where:

c: {1:61:12:11:1:145}

p: {1:5:12:7:5:29}

(29-5) = 2*12.

Perhaps there is a way to solve our existing equations based on this new relationship between a, b and 2d.

Also, in the attached picture, notice there is a second record that satisfies this equation that we can get to quite easily from c.

At (1,785, -378). This is the negative x for a record created at (e, c, t=c.t+c)

>>2975

Little square, confirmed. Not seeing it yet…

>>2977

Looks like (d+1)^2 - d^2 == f + 1.

It's a pity the small square formulas in terms of t only work at (e,1).

>>2991

welcome back. Any more crumbs?

>>2990

Thank you. Didn't notice this post earlier.

>>2974

Just a quick update, the a[t]=na record also satisfies b-a = 2D.

So all that searching, and we had a valid starting position all along.

>>2997

>e is the remainder of the left, and f is the remainder of the right.

Much appreciate this clear explanation.

>>3012

>>3015

Not sure if I'm understanding this hint either. I was under the impression we were looking for a matching x or matching t value going backwards from na.

Perhaps we are looking for matching calculations higher up.

See pic attached for c=145.

>>3017

>It's right there! x = 7 and a = na!

Sure. But without searching, how do we know that's the correct record?

>>3019

VA - that's our standard (n-1)*a transform from (e.n) to (e,1). Our starting position.

Here's another example for c=901.

Only showing records where d[t]-d mod n-1 or a[t] mod n up to t=c.

>>3021

I'm running for all my test cases. Not seeing anything obvious here.

>>3024

No problem. See attached.

Also added a property at the end "dadiff". Represents diff between the d[t] calc and the a/n calc. Looking to either match the solution p.x, or the diff between c.x and p.x.

Last one at c=785.

Interesting record at (1,1,379).

Diff calculation is 4, same as diff in x.

>>3028

Yes. Any (e,n) can be transformed to (e,1). The (1,1,4) record where x=7 and a=25 is the na transform of the prime solution for c=145.

However, we still don't have any calculation to navigate up the tree and determine if that is the solution record. We "know" it's correct. We just can't prove it.

>>3030

starting record at c=145

(1,61,6) = {1:61:12:11:1:145}

na transform to (1,1,6)

prime solution at a=5,b=29

(1,5,4) = {1:5:12:7:5:29}

na transform to (1,1,4)

Shortcut for the na transform is to take any record at (e,n,t) and create another record at (e,1,t). The a value for that record will be n*a from the original with n = 1.

Still exploring d[t] formulas trying to find a link to any prime related record.

Trusty c=145 example in pic.

Records are filtered to show only valid factors of the d[t] and a[t] formulas.

I've added a column to test

( ( (d[t] - d) / (n-1) ) - e ) / d

to see if any records would stand out as more relevant. Interestingly, in all my test cases, (x^2 - f) - e is always a valid factor of d.

Still not sure if I'm heading down a valid path.

>>3149

>Maybe something in (n, 0) will yield something interesting.

See - welcome back. There was a discussion about this in the previous thread.

Not sure if the following is helpful.

>>2013

>>2016

I checked the original grid and the n=0 records do have some special properties:

e is negative and one of = 1^2, 2^2, 3^2, 4^2, etc.

x = sqrt(-e)

b-a = 2x

d-b = x

BigInteger newE = -(ter.d * ter.d);

BigInteger newX = ter.d;

BigInteger newA = ter.n;

TheEndRecord testNegative = TerFactory.CreateForEXA( newE, newX, newA );

>>3159

Can you share that formula for d?

>>3172

No problem. I've found a way to iterate values in (-e, 0).

Have you noticed that records exist matching the prime solutions?

c: (1,25,4) = {1:25:8:7:1:65} = 65

p: (1,1,2) = {1:1:8:3:5:13} = 65

(-16,0,3) = {-16:0:9:4:5:13} = 65

c: (1,61,6) = {1:61:12:11:1:145} = 145

p: (1,5,4) = {1:5:12:7:5:29} = 145

(-144,0,7) = {-144:0:17:12:5:29} = 145

c: (249,10065,71) = {249:10065:142:141:1:20413} = 20413

p: (249,1,3) = {249:1:142:5:137:149} = 20413

(-36,0,4) = {-36:0:143:6:137:149} = 20413

>>3175

Thank you.

Here's a very sloppy method for generating an (-e, 0) record for any other record. Haven't fully debugged as to why I can't create the record from (e,n,d,x,a,b) variables, but the e,x,a are certainly valid.

public static TheEndRecord CreateForZeroN( TheEndRecord ter ) {

BigInteger a = ter.a;

BigInteger b = ter.b;

BigInteger n = 0;

BigInteger x = (b - a) / 2;

BigInteger d = x + b;

BigInteger e = -(x * x);

return CreateForEXA( e, x, a );

}

Regarding the "mirror", it's a bit confusing.

We can get to the negative e side of the grid with e - ((2 *d) + 1).

Then there is the -x space at (e,n) where a and b values reverse.

And now there is the -e, 0 space where we can find matching prime results.

They all play a role.

>>3180

(shocking)

>>3179

These are some methods I put together a while ago that mostly worked for any (e,n). I say mostly, because Teach found a case in (56,n) I believe that didn't quite fit.

Also, I think we also discovered the ability to iterate records by both (x+n) and (x-n) adjustments to d after this. But should be a reasonable starting position.

public static TheEndRecord GetNext( TheEndRecord ter ) {

BigInteger newd = GetNextD( ter );

return TerFactory.CreateForEND( ter.e, ter.n, newd );

}

private static BigInteger GetNextD( TheEndRecord ter ) {

if ( ter.n % 2 == 0 ) { // even

return ter.d + (ter.x + ter.n) + ter.n / 2;

} else { // odd

return ter.d + 2 * (ter.x + ter.n) + 2 * ter.n;

}

}

public static TheEndRecord GetPrevious( TheEndRecord ter ) {

BigInteger newd = GetPreviousD( ter );

return TerFactory.CreateForEND( ter.e, ter.n, newd );

}

private static BigInteger GetPreviousD( TheEndRecord ter ) {

if ( ter.n % 2 == 0 ) { // even

return ter.d - (ter.x + ter.n) + ter.n / 2;

} else { // odd

// to walk down the tree: new.d = orig.d - 2*(x+n) - 2n

// can also be written as ter.d - 2 * ter.x

return ter.d - 2 * (ter.x + ter.n) + 2 * ter.n;

}

}