Very cool, PMA!
This part is cool! It also matches an earlier PMA diagram!
>each of the eight triangles will have one OR one of two (the latter when c is large and the product of two different prime numbers) configurations in each triangle. The difference between those two configurations of a portion of f are that they are both staircase numbers where the base of one is a unit longer than the other. E.g. (3,4,5) and (4,5,6)
Here's a quick draw. Each inner rectangle has dimensions of u by u+1. This is just for visualizing the idea, and for small n it would be different. But I think for large n the pieces of the squares are ordered correctly. Thoughts, Anons?
Alright guys, I think I just found something really cool. Need your help to verify Pls!
So, for c = 6107
f = 134
f-2 = 132
(f-2)/8 = 16
(f-2) mod 8 = 6
Based on VQC's latest hints, there are two staircase numbers that make up (f-2)/8 which give the correct n0 inside each one of the 8Tu. I think I found out how to find them:
4+3+2+1 = 10
3+2+1 = 6
= 16
The correct f shape inside each of the 8Tu could be a triangle?
Key idea is that the overlay of (1+2+3) and (1+2+3+4) = 16
Overlay doesn't work for filling the square.
But making a triangle along the straight side of each Tu works. Thoughts, Anons?
Thanks PMA! More verification needed, but hopefully this turns out to be a good step forward. Can we find the two triangular numbers that create (f-2)/8 ? Is that the formula you're talking about?
CA, check out row 1!
Sqrt(f) is ALWAYS (x+n)
f is also (x+n)^2 always.
That's why (e,1) row 1 is our reference to the correct solution.
Somehow row 1 gives us a clue to the (prime) solution.
Thoughts, Anons?
Good explanation, MA.
Hello VQC, nice to wake up on a Saturday and see your posts! Studying now. The adjacent polite triangle numbers are the key. We're working on how to make the remainders and "leftovers" fit. I think that's what Topol's new drawings are hinting at. Thanks for the new crumbs, back to work!
Trying to work this out in a real example.
c=6107
f=134
(f-2)=132
(f-2)/8=16
(f-2) mod 8 = 4
(x+n) = 83
(d+n) = 114
u1 = 41
u2 (u+1) =42
correct n value = 36
Here's the formula: nn + 2d(n-1) + f - 1
f-1 = 134-1 = 133
2d(n-1) = 2*78(36-1) = 5460
nn = 1296
1296 + 5460 + 133 = 6889
SQRT(6889) = 83 = correct (x+n)
Thoughts, Anons?
Nice digits, AA! I'm just building out the c=6107 example to try and complete a medium sized problem IRL. I'll post updates, trying to fit everything together like a block puzzle.
Here's layer 4.
c=6107
f=134
(f-2)=132
(f-2)/8 = 16 (green triangles = 6 + yellow triangles =10)
(f-2) mod 8 = 4 (blue squares)
Current Values for layer 4:
(f-2)/8 value = 128 18 = 2304 = 24 * 24 4
(f-2) mod = 4 * 18 = 72
2304 + 72 = 2376
Check 132 * 18 = 2376,
mods + (f-2) values currently matching.
I think the mods are going to form a new border to create the +1 differential need for u and u+1.
Ok, lads. I have a match. Still don't quite understand how it all works. But additional iterations of 16 filled the gap. Take a look, can't confirm it's 100% correct, but the symmetry / asymmetry is pretty cool.
The border needed corners and midpoints of 40/8=5. So each corner and midpoint has and odd number. The 8 remaining gaps to be filled were all 36 units. Found a way to make them all fit.
Yellow = 6
Green = 10
They all fit.
But shouldn't there be extra mod units to add in? I'm stoked but at the same time confused. Any ideas welcome. Posting a work in progress for group discussion.
Thanks CA!
Also, every Anon archive everything offline. I've got most of it, but need to save the maps first. Baker did a great job with that.