>{40:220:21:20:1:481} (40, 220, 11)
Ok, so transforming this to N=0, the new record is:
{-57600:0:241:240:1:481}
using:
new e = 40-221220-220*220
new d = orig d + change in n, or 21+220=241
new x = orig x + change in n, or 20+220=240
So then, relation of:
{-57600:0:241:240:1:481}
to
{-144:0:25:12:13:37}
is… the next MisterE?
>…for each extension of these variables you'd need a separate dimension of the grid, though,…
Have been seeing this as well. Appears the additional 'dimensions' are related to derivatives (rates of change, how the variables GROW), or second derivatives, sometimes interleaved. These are associated with the integer 'points' in the grid.
Gut and some research seems to be pointing toward a Clifford Algebra associated or overlayed on the grid. When we have those vectors for two points of interest, some inner product or other transformation would lead to a solution. Have been interested in learning about Clifford Algebra for a while as a foray into next level math understanding. Also, there were some suggestions earlier by Chris to dig into in the meantime, and the digging keeps turning up Clifford Algebra (will have to sauce that later).
Baker, we're over 700 posts. Can we consider including Cliff in the next bread, to complement Fermat in RSA#8? Some interesting stories, twists and turns in mathematics / physics, resurgence in his work decades later, and likely has a bearing on where we're headed. Touches on roots, modular number systems, quaternions, etc. It gets at the geometrization
of the real number system. Plus he wore an epic beard.
Some quotes from William Kingdon Clifford (they read like an anon credo!):
https:/ /libquotes.com/william-kingdon-clifford
Checked!
Thanks anon, this was an interesting dive, only part way through understanding (haven't dug into patterns at n=1).
>But how is this a solution when the a and b values are still unknown?
>What am I missing?
Don't think you're missing anything, and not sure yet either, but useful for a quick jump to n=0 or n=1.
It also produces a new e that is a perfect square. The new t is always an integer that is the square root of the negative of the new e. This is expected because we know:
e=2an-x^2, and if n=0, e=-x^2
I think being able to transform to a perfect square column in the -e space may be useful once everything is worked out. There is a subsequent transform of 'a' needed, as illustrated below.
Attached images relevant.
First is the orig VQC, generated by Teach back in Dec and on pastebin. Showing only the -e perfect square columns, for n=0 and n=1. The gray bar is the first record, passing through {0:0:1:0:1:1}. Added x's starting at e=-9 above bar, as a sequence for 'a' could be extended upward with decreasing a values (moving t through zero to negative?).
Note: at n=0: d, a, and b increment by 1 for each t. x never changes. x^2=-e. There is a gap between a and b, which is 2*x, and this never changes for t sequences. Also, d-x=a for any n=0.
Note: observe how d increases as we move leftward decreasing e (each +1 increase in x). It's not linear.
seq:1,2,3,5,9,13,19,25,33
See: https:/ /oeis.org/A085913
A085913: Group the natural numbers such that the product of the terms of the n-th group is divisible by n!. (1),(2),(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),… Sequence contains the first term of every group.
1, 2, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405
There is a similar (same initially) sequence
See: https:/ /oeis.org/A099392
A099392: a(n) = floor((n^2-2*n+3)/2).
1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405
The second formula was easier in excel, but it breaks down at x=54. The d, a, b, and c get out of sync, and no longer check out (see highlighted region in screen cap). Maybe that's why the oeis.org sequence ends there? Maybe the A085913 sequence based on the factorial and groups would continue to work? Not sure, but this doesn't scale at this point for d, using this quick excel model.
also highlighted in that image is e=-1849, the transform for 87 (1x87) to n=0: {-1849:0:44:43:1:87}, if anyone wants to find a link to the 87 (3x29) record transformed to n=0: {-169:0:16:13:3:29}
Ok, here are transformation examples with a few of the 'known' factors we've been using. Doesn't apply to getting from the 1c to desired ab factors yet, but interesting patterns. Refer to image with A085913 in top left, starting at e=0.
Take 85 (5x17)
{4:2:9:4:5:17}
transformed to n=0 is:
{-36:0:11:6:5:17}
Calc d for t=1 at x=6 is 19 (highlighted tan), and we want d=11, so transform 8 steps. a and b are 13 and 25, so taking away 8 from each is 5 and 17, 5x17=85, check.
87 (3x29)
{6:7:9:6:3:29}
to
{-169:0:16:13:3:29}
Calc d for t=1 at x=13 is 85 (highlighted tan), and we want d=16, so transform 69 steps. a and b are 72 and 98, so taking away 69 from each is 3 and 29, 3x29=87, check.
145 (5x29)
{1:5:12:7:5:29}
to
{-144:0:17:12:5:29}
Calc d for t=1 at d=12 is 73 (highlighted tan), and we want d=17, so transform 56 steps. a and b are 61 and 85, so taking away 56 from each is 5 and 29, 5x29=145, check.
Looks very interesting CA! Will have to dig into that when some time frees up, maybe you've got this!!!
Have been enjoying the patterns you point out Isee. Those are indeed interesting.
also, some of the patterns PMA has pointed out with delta d's, and interleaved second differentials I see as well.
for example, (1:5) attached pic.
I'll throw a post together with some other patterns done visually.
Also, check out the interesting patterns when visualizing Fermat's last theorem, screen capped from a tube. I think this is what we have going on in a way.
>… LOOK AT HOW CLOSE TO WINNING WE ARE!
Ok, but how do you KNOW we are close to winning?? Anything more to share?
> Reread Chris crumbs!
Yes, still working through them, picked this up a couple weeks ago to actually work through stuff (where Teach and all were at mid Dec). Each crumb leads to a deeper understanding of the VQC/GRID. One of the things that keeps me going on this, LARP or not, is that Chris has always been coherent in his crumbs. Granted, we could start dropping some crumbs with the grid now, even without an end solution, but it sides more with an 'informed faith' rather than a 'blind faith' at this point.
we just need to convince Chris that a) we're ready for the next walk-through tutorial session, and b) it's Time for that Now to get ready for next phase of bigger picture! (hint hint Chris).
Ok, here are a few playful images / exploration of patterns. Follows along with the fermat visualization above.
First, I took the orig grid output, e-64 to e64, t=1 to 10, n=0 to 63.
Zoomed out, you can see patterns right away.
Added darker shading to the cells with values. Some ranges only have a few t's, as the d, a or b capped out and stopped generation, but they would go on.
Wouldn't quite print right, so generated a png output for the left side and right side of the grid.
Could play in an image program, but chose to print these out, and play on a light table.
I see 2 parts to our goal: 1) be able to hop/transform to another cell that is valid (not empty), and b) able to go in proper direction(s) and transform appropriately along the way. Ala hopping the lilypads on the grid-pond.
That's cool CA, love your images!
Yes, that's exactly the pattern.
Also trying to look at the shifting / reflecting / twisting perspective.
Attached are the first set, looking at +e. Printed 2 copies and overlayed on light table. Then shifted. The darker cells are the ones where you would be able to jump and land on a cell (valid transform or not).
The perspective view makes the angled lines jump out. The strongest at n=0, e=0, which is probably the Root O D. These make sense with the math as well, including the slope (e can increase by 2 for each increase in one unit).
Ok, to finish off these playful patterns.
Looking at just the Negative e and shifting. These show the strongest lines, with a primary line slope of 2 running away from each perfect square at the n=0 origin. Other lines run different direction, opposite slope, but not 2.
Next reflecting along hortizontal axis (along an n) with the -e space. What is interesting about this, is that each perfect square e column has no gaps, meaning, you could jump from n to -n, back and forth. I think this would mean move along a perfect square column -e toward n=0, but, if you run into a gap, then skip over to the other side (plus n), and carry on toward n=0 until a gap, then jump back, and so on.
Finally a little rotation for fun - the VQC Vortex.
>>3348 Nice!
Interesting, the t diff binary pattern really jumps out visually.
Also reminds me of some patterns from a while back, think it was you?
Just that the a for an n, plus x, plus a for n+1 is the new a (at n+1), the a for n+1 is b for n. This looping pattern that cascades down.
Another pattern is the final digits for a, b, and c. For (2,1), c always ends in a 3, 7 or 1 digit.
For (1,1), it's always a 5 or a 1 for c. Similar patterns in a and b.
Reminds me to looking into modular forms for our a's, b's and c's for patterns and what they say about the rates of change, or about the factors / relationships.
> Also the D values are the same as the A values for the (e-1,1) cell.
-Be careful, looks as if that only holds for initial e that are even. The D is same as the (e-1, 1, t+1) A if starting e is odd.
But, in the orig VQC output, this isn't always consistent in the negative e space, there are a few odds that are the same. Might be an artifact of where t=1 started and incorrect grid production (see attached image, e=-39 to e=-40 isn't same pattern as others).
And by extension, the D values are the same as the B values for the (e-1, 1, t-1) cell if e is even, and the same for the (e-1, 1, t) cell if e is odd.
Interestingly, this pattern holds in the negative e space for n=0, if jumping from perfect square to square (-9 to -16, -36 to -49), as shown in first image in previous post here >>3344. Need to validate further, but looks consistent.
Thanks Baker, new bread looks superb, looking forward to the first slice.
This is the pdf I use first for crumbs, nice compilation, have on phone. Probably everything needed is in there to unlock this with the (n)KEY.
Have been catching up on the crumbs, starting a couple weeks back, still a month behind. I need to dive into p next.
Hey VA, good to read you! Maybe a nice UK graphic will pull V in here.
CA, I recall your other D grid here >>2890 that was interesting.
>Regarding interwoven / interleaved patterns, you'll see theres 4 or 5 I think in (0, 8). 2 isn't a magic number, I suspect that you can find infinite interwoven patterns in any (e, n) if you extend the grid into infinity.
>Maybe there is a limit, but I haven't spent any time looking into it…
Hey Isee, seem to recall Chris specifically saying there are a limited number of patterns, and we should enumerate them (how they GROW). Can't find the crumb right now, but think this is critical. I see a 'pattern' as a 1st, 2nd… order rate of growth (derivatives, you know we're going to end up doing some sort of geometric calculus anons!) for each of our variables, and may include interleaved patterns. Believe we need to be able to extract/recognize/model those patterns from the relevant n=1 row elements, and apply the pattern to the relevant base values to get the jump(s)/shift(s) we need. We will benefit from using an appropriate CHOSEN prime number, multiplied by our c, to help/enable the transformation / ID of factors.
Noted your D-sequence is the same I listed earlier here >>3344
Formula is: a(n) = floor((n^2-2*n+3)/2)
in Excel: =FLOOR.MATH(((E10+1)^2-2*(E10+1)+3)/2)
where E is an 'x' value, these increment by one. But as I mention, appears to break down later after the 53rd x. Perhaps the other method would work, or if you could share more details on how you're generating that would be appreciated.
Last bit for today, is a look at n=0, and entering c-values into excel (doing manually still).
This gives a visual for how the "e space" between D values grows over time (+2 for each incremental D).
First column is the D value, sequence of integers. Second column indicates if D is prime (have a lookup table with first 100K primes that goes up to 1.2M). Then D^2, and then diff between squares of D floor and D ceiling. For each increase in D, +2 is added to the e value range.
So for our 145 example, between d of 12 and 13, the gap is 25 (max e, think one less than that actually, need to ignore perfect squares on either side where e=0), and actual e is 24 to reach 169. The 145 input, gap, and e are in the top row.
Another example shown, c=12,657,734,632, e=90,417. The e range up here is 225,013. Still small compared to RSA, but enough to play once all is figured out.
Hey VA, if you want to input some numbers to hit that home run, here they are (from Chris), it would break my excel model though!
public static string Rsa100c = "15226050279225333605356183781326374297180681149613806886579084945801229632589528976540003506920061 39"; public static string Rsa100a = "37975227936943673922808872755445627854565536638199"; public static string Rsa100b = "40094690950920881030683735292761468389214899724061"; public static string Rsa100d = "39020571855401265512289573339484371018905006900194"; public static string Rsa100e = "61218444075812733697456051513875809617598014768503"; public static string Rsa100f = "16822699634989797327123095165092932420211999031886";//2d+1-e public static string Rsa100n = "14387588531011964456730684619177102985211280936"; public static string Rsa100x = "1045343918457591589480700584038743164339470261995"; public static string Rsa100x_plus_n = "1059731506988603553937431268657920267324681542931";