Katabatak Math-An Exploration In Pure Number Theory

[Turtle]

___Welcome once again expeditioners! Congratulations for coming this far! All's the more credit for you who choose to continue in view of the infinite scope of this adventure.
___Now that a few have struck out on their own I will try and hold my own observances a bit until they return with their discoveries. Indeed I have not yet enetered the caverns of infinite series/sequences which first brought me in & that will come soon enough.
___In the mean time I see it's time for another of those gatherings; help yourselves to what you need & make yourself comfortable among like minded folk. It may not be too far off that we may add Katabatak music to the victuals, libations, & dancing. Carpe diam!

[Turtle]

___In the mean time we need to reference the skew table in post #38 & the Base Ten Katabatak Addition table in post #44. First, to the algebraic notion preceding each Katabatak class in the skew table. For class 1 the expression is 9n-8, class 2 is 9n-7, class 3 9n-6, etc.. In the expressions, n is the ordinal number from the class set; so the 3rd element of the class 1 is (9*3) -8 = 27 -8 = 19.
___In passing, I note that 9n+1 is equivalent to 9n-8, but requires the ordinal index to begin with 0 (zero) & that is precisely what we don't want.
___So, replace the list & table elements in post #44 with algebraic notaions & stir by class as follows.


______ K Class 1 ________/______ K Class 2 ____________/
1 + 9 = 1________________/_1 + 1 = 2_________________/
(9m-8) + (9n) = (9p-8)_____/_(9m-8) + (9n-8) = (9p-7)____/
9m+9n-8 = 9p-8___________/_9m+9n-16 = 9p-7_________/
9m+9n = 9p_______________/_9m+9n-9 = 9p___________/
m + n = p________________/_m + n - 1 = p____________/
_________________________/________________________/
2 + 8 = 1________________/_2 + 9 = 2_________________/
(9m-7) + (9n-1) = (9p-8)___/_(9m-7) + (9n) = (9p-7)______/
9m+9n-8 = 9p-8___________/_9m+9n-7 = 9p-7___________/
9m+9n = 9p_______________/_9m+9n = 9p______________/
m + n = p________________/_m + n = p________________/
_________________________/_________________________/
3 + 7 = 1________________/_3 + 8 = 2_________________/
(9m-6) + (9n-2) = (9p-8)___/_(9m-6) + (9n-1) = (9p-7)____/
9m+9n-8 = 9p-8___________/_9m+9n-7 = 9p-7__________/
9m+9n = 9p_______________/_9m+9n = 9p_____________/
m + n = p________________/_m + n = p________________/
_________________________/_________________________/
4 + 6 = 1________________/_4 + 7 = 2_________________/
(9m-5) + (9n-3) = (9p-8)___/_(9m-5) + (9n-2) = (9p-7)____/
9m+9n-8 = 9p-8___________/_9m+9n-7 = 9p-7__________/
9m+9n = 9p_______________/_9m+9n = 9p_____________/
m + n = p________________/_m + n = p________________/
_________________________/________________________/
5 + 5 = 1________________/_5 + 6 = 2________________/
(9m-4) + (9n-4) = (9p-8)___/_(9m-4) + (9n-3) = (9p-7)___/
9m+9n-8 = 9p-8___________/_9m+9n-7 = 9p-7__________/
9m+9n = 9p_______________/_9m+9n = 9p_____________/
m + n = p________________/_m + n = p________________/

[Turtle]

___The algebraic model of the Katabatak addition described above has indecies m and n, where m and n have integer values > 0. The first column, class of K(1), lists the 5 possible equations with their derivation, and not listed but pertinent we keep in mind their reflexive partners. That is for example in the first entry, 1 + 9 = 1 is listed, but not 9 + 1 = 1.
___Here is where the primary graph with seperation finds utility; black squares constitute the Katabatak class one elements.

[Turtle]

___Don't take the view that I understand these structures completely; I just know how to draw them to the attention. For example, all of the expressions/equations resolve to one of 2 forms, either m + n = p or m + n - 1 = p & I have no firm idea yet how to understand this in regard to the primary graphs. Take your time & enjoy the tour & don't hesitate to accomodate our tour host Tormod, without whom few but myself gain the opportunity to explore here.

______ K Class 3 ________/______ K Class 4 _________/

1 + 2 = 3_________________/_1 + 3 = 4________________
(9m-8) + (9n-7) = (9p-6)____/_(9m-8) + (9n-6) = (9p-5)_
9m+9n-15 = 9p-6__________/_9m+9n-14 = 9p-5__________
9m+9n - 9 = 9p___________/_9m+9n-9 = 9p_____________
m + n -1 = p_____________/_m + n - 1 = p____________
_________________________/_______________________
3 + 9 = 3_________________/_2 + 2 = 4________________
(9m-6) + (9n) = (9p-6)_____/_(9m-7) + (9n-7) = (9p-5)_
9m+9n-8 = 9p-8___________/_9m+9n-14 = 9p-5__________
9m+9n = 9p_______________/_9m+9n - 9 = 9p___________
m + n = p________________/_m + n -1 = p_____________
_________________________/________________________
4 + 8 = 3_________________/_4 + 9 = 4________________
(9m-5) + (9n-1) = (9p-6)____/_(9m-5) + (9n) = (9p-5)___
9m+9n-6 = 9p-6___________/_9m+9n-5 = 9p-5_________
9m+9n = 9p_______________/_9m+9n = 9p____________
m + n = p________________/_m + n = p_______________
_________________________/________________________
5 + 7 = 3_________________/_5 + 8 = 4_______________
(9m-4) + (9n-2) = (9p-6)____/_(9m-4) + (9n-1) = (9p-5)_
9m+9n-6 = 9p-6___________/_9m+9n-5 = 9p-5_________
9m+9n = 9p_______________/_9m+9n = 9p____________
m + n = p________________/_m + n = p_______________
_________________________/________________________
6 + 6 = 3_________________/_6 + 7 = 4________________
(9m-3) + (9n-3) = (9p-6)____/_(9m-3) + (9n-2) = (9p-5)_
9m+9n-6 = 9p-6___________/_9m+9n-5 = 9p-5___________
9m+9n = 9p_______________/_9m+9n = 9p______________
m + n = p________________/_m + n = p________________
_________________________/_________________________

______ K Class 5 ________/______ K Class 6 _________

1 + 4 = 5_________________/_1 + 5 = 6______________
(9m-8) + (9n-5) = (9p-4)____/_(9m-8) + (9n-4) = (9p-3)_
9m+9n-13 = 9p-4__________/_9m+9n-12 = 9p-3_______
9m+9n - 9 = 9p___________/_9m+9n-9 = 9p__________
m + n -1 = p_____________/_m + n - 1 = p___________
_________________________/______________________
2 + 3 = 5_________________/_2 + 4 = 6______________
(9m-7) + (9n-6) = (9p-4)____/_(9m-7) + (9n-5) = (9p-3)_
9m+9n-13 = 9p-4__________/_9m+9n-12 = 9p-3_______
9m+9n - 9 = 9p___________/_9m+9n - 9 = 9p_________
m + n -1 = p_____________/_m + n -1 = p____________
________________________/_______________________
5 + 9 = 5_________________/_3 + 3 = 6______________
(9m-4) + (9n) = (9p-4)______/_(9m-6) + (9n-6) = (9p-3)_
9m+9n-4 = 9p-4___________/_9m+9n-12 = 9p-3________
9m+9n = 9p_______________/_9m+9n -1 = 9p__________
m + n = p_________________/_m + n -1 = p____________
_________________________/________________________
6 + 8 = 5_________________/_6 + 9 = 6________________
(9m-3) + (9n-1) = (9p-4)____/_(9m-3) + (9n) = (9p-3)___
9m+9n-4 = 9p-4___________/_9m+9n-3 = 9p-3___________
9m+9n = 9p_______________/_9m+9n = 9p______________
m + n = p________________/_m + n = p________________
_________________________/_________________________
7 + 7 = 5_________________/_7 + 8 = 6________________
(9m-2) + (9n-2) = (9p-4)____/_(9m-2) + (9n-1) = (9p-3)_
9m+9n-4 = 9p-4___________/_9m+9n-3 = 9p-3__________
9m+9n = 9p_______________/_9m+9n = 9p_____________
m + n = p________________/_m + n = p________________
_________________________/_________________________



______ K Class 7 ________/______ K Class 8 _________

1 + 6 = 7________________/_1 + 7 = 8______________
(9m-8) + (9n-3) = (9p-2)_/_(9m-8) + (9n-2) = (9p-1)_
9m+9n-11 = 9p-2__________/_9m+9n-10 = 9p-1_______
9m+9n - 9 = 9p___________/_9m+9n-9 = 9p__________
m + n -1 = p_____________/_m + n - 1 = p___________
_________________________/______________________
2 + 5 = 7________________/_2 + 6 = 8_______________
(9m-7) + (9n-4) = (9p-2)_/_(9m-7) + (9n-3) = (9p-1)_
9m+9n-11 = 9p-2__________/_9m+9n-10 = 9p-1_______
9m+9n - 9 = 9p___________/_9m+9n - 9 = 9p_________
m + n -1 = p_____________/_m + n -1 = p____________
_________________________/______________________
3 + 4 = 7________________/_3 + 5 = 8_______________
(9m-6) + (9n-5) = (9p-2)_/_(9m-6) + (9n-4) = (9p-1)_
9m+9n-11 = 9p-2__________/_9m+9n-10 = 9p-1_______
9m+9n-9 = 9p_____________/_9m+9n -1 = 9p_________
m + n -1 = p_____________/_m + n -1 = p____________
_________________________/______________________
7 + 9 = 7________________/_4 + 4 = 8_______________
(9m-2) + (9n) = (9p-2)___/_(9m-5) + (9n-5) = (9p-1)_
9m+9n-2 = 9p-2___________/_9m+9n-10 = 9p-1_______
9m+9n = 9p_______________/_9m+9n-9n = 9p________
m + n = p________________/_m + n -1 = p___________
_________________________/______________________
8 + 8 = 7________________/_8 + 9 = 8_______________
(9m-1) + (9n-1) = (9p-2)_/_(9m-1) + (9n) = (9p-1)___
9m+9n-2 = 9p-2___________/_9m+9n-1 = 9p-1________
9m+9n = 9p_______________/_9m+9n = 9p___________
m + n = p________________/_m + n = p______________
_________________________/_______________________

______ K Class 9 ________/

1 + 8 = 9________________/
(9m-8) + (9n-1) = 9p_____/
9m+9n-9 = 9p_____________/
m + n -1 = p_____________/
_________________________/
2 + 7 = 9________________/
(9m-7) + (9n-2) = 9p_____/
9m+9n-9 = 9p_____________/
m + n -1 = p_____________/
_________________________/
3 + 6 = 9________________/
(9m-6) + (9n-3) = 9p_____/
9m+9n-9 = 9p_____________/
m + n -1 = p_____________/
_________________________/
4 + 5 = 9________________/
(9m-5) + (9n-4) = 9p_____/
9m+9n-9 = 9p_____________/
m + n -1 = p_____________/
_________________________/
9 + 9 = 9________________/
(9m) + (9n) = (9p)_______/
m + n = p________________/
_________________________/

[Turtle]

__It looks like we (I) recquire an understanding of what the value p means in regard to the primary graph. I notice that those expressions resolving to m+n-1=p, plot graph cells all in the upper left diagonal half of the graph.
___Having a better view of this may help facilitate further advance on the proofs of Fermat's last theorem I showed earlier. We camp here!


erratum: Since the graph has 81 cells (9x9), there is no integral diagonal half in terms of cells. The noted region has 36 cells.

[Turtle]

___Forgive my absense; I found myself stuck upside down after trying to take the proper perspective on this problem of Fermat's. Now that I have regained my feet, I see my difficulty lay not in the view I took, but rather that I rushed to it.
___I intend to go by way of the algebraic equivalences of the Katabatatak classes, so here again the base ten skewed table of the integers by Katabatak classes.

___Below find a neatly typed list of the base ten Katabatak additions which generate class 1 elements.
___Following the 2 together, see that if you add the m'th K(1) to the n'th K(9), the result is the p'th K(1). Let m & n = 1 then the first K(1) [1] plus the first K(9) [9] equals the second K(1) [10] & substituting in the expressions we have ((9*1) -8)+(9*1)=9p-8 then 9-8+9=9p-8 then 18=9p and so p=2. Going to the skew table, find the 2'nd element in the row of class K(1) [10].
___Hope this doesn't invert anyone! Make haste slowly.


______Class 1 _________
1 + 9 = 1
(9m-8) + (9n) = (9p-8)
9m+9n-8 = 9p-8
9m+9n = 9p
m + n = p

2 + 8 = 1
(9m-7) + (9n-1) = (9p-8)
9m+9n-8 = 9p-8
9m+9n = 9p
m + n = p

3 + 7 = 1
(9m-6) + (9n-2) = (9p-8)
9m+9n-8 = 9p-8
9m+9n = 9p
m + n = p

4 + 6 = 1
(9m-5) + (9n-3) = (9p-8)
9m+9n-8 = 9p-8
9m+9n = 9p
m + n = p

5 + 5 = 1
(9m-4) + (9n-4) = (9p-8)
9m+9n-8 = 9p-8
9m+9n = 9p
m + n = p

[Turtle]

___If any one is still standing, you may want to go ahead & sit; I may start a little shaking here.
___First step, the attachment below which is the base ten Katabatak addition table stripped of its indices & seperations. Review the key: {1-black,2-red,3-orange,4-yellow,5-green,6-blue,7-indigo,8-violet,9-white}
___Now, pick any cell as a starting point & note the value; to add, move either right or down on the table the number of cells you wish to add (if you reach an edge, skip back to the top or left in the same column or row). The cell you stop in is the solution. By using a start & making two additions, & then using the same values but starting with the middle value, the associative principle is demonstrated.
___I noticed that even though I provided a table for subtraction earlier, it isn't necessary. Simply proceed as above except move from any start either left or up.
___Finally, albeit a stretch to visualize , the cropped table wraps nicely on a torus thus negating the edges. I love camping!

[Turtle]

___But wait!; I see more! Moving diagonally down & right is equivalent to adding twos, while moving diagonally up & left is subtracting twos; moving diagonally up & right is equivalent to adding nines, while moving diagonally down & left is subtracting nines. (Remember 9 is the base ten Katabatak additive identity.) Topologist? Is there a topologist in the house? Someone throw me a lifering! I love no edges camping.

[Turtle]

___Camping takes on its own brand of organizational economy which relies on reusability. Just so, I have started reusing post #98 to list the typed algebraic equivalencies as I finish them.

[Turtle]

___On any extended expedition of exploration, camping is inseperable from going place to place. Some of the goings-on of camp stay the same whether many or just one camp; preparing food & cleaning utensils, setting up shelter, collating a field journal, marking samples, planning the next movements, etc.. When more than one camps, these same concerns apply now subject to discussion. A different set of concerns now emerges in a social context; concerns such as division of labor, story telling, & points of contention.
___Anyway, pull up a log & enjoy the camp.
___To restate where I plan to go briefly, I want to find more neat little Katabatak demonstrations of special cases of Fermat's last theorem. Here in camp I like playing with the base ten Katabatak addition skew-table/torus. If you can, imagine it painted on one of those liquid-filled torus toys that squirts out of your hands every time you seize ahold of it; now slip it on a dowel snug & you have a Katabatak residue calculator. Roll the torus down sequentialy to add & up to subtract; rotate the dowel clockwise to add & counter-clockwise to subtract.
___Whether you like the torus or the table in post, I mention in passing it multiplies & involutes Katabatak residues as well.