Katabatak Math-An Exploration In Pure Number Theory

[CraigD]

… Once that door was opened to my growing mind I soon deducted the true formula for integer right triangles was a^2 = 2b + 1, where a is a positive integer greater than 1.

Thank you, Bill, for this very thought (and algebra) provoking post!

 

I haven’t had a chance to fully read your observations apply the k(n) to the sum of the sides of the base triangles in the formula, but your formula

a^2 = 2b +1, where a is not a multiple of by 2

is true (the proof is not too difficult, especially if put in terms of the numeration scheme I describe in “PRMPF - generating the prime factorizations of the integers without using division” :rainbow:)

 

It's a special case of N=1 of a general formula

a^2 = n(2b+n), where a is not a multiple of n+1 (actually, something a bit more complicated I can't seem to find)

 

For example, here are some right triangles for n=1 to 25: 3x4x5, 5x12x13, 7x24x25, 9x40x41, 11x60x61, 13x84x85, 8x15x17, 12x35x37, 16x63x65, 15x20x25, 21x28x35, 35x84x91, 20x21x29, 28x45x53, 36x77x85, 33x56x65, 39x80x89, 33x44x55, 39x52x65, 51x68x85, 48x55x73, 57x76x95, 65x72x97

 

I never cease to be amazed at how almost all observations about integers (or rational numbers) seem to follow these formulae. It’s all pointing at, I believe, a fundamental “calculus of prime numbers” yet be to discovered!

 

PS: I've started a new thread 5179, to explore the non-katabatak aspects of this formula.

[TheBigDog]

For example, here are some right triangles for n=1 to 25: 3x4x5, 5x12x13, 7x24x25, 9x40x41, 11x60x61, 13x84x85, 8x15x17, 12x35x37, 16x63x65, 15x20x25, 21x28x35, 35x84x91, 20x21x29, 28x45x53, 36x77x85, 33x56x65, 39x80x89, 33x44x55, 39x52x65, 51x68x85, 48x55x73, 57x76x95, 65x72x97

 

I never cease to be amazed at how almost all observations about integers (or rational numbers) seem to follow these formulae. It’s all pointing at, I believe, a fundamental “calculus of prime numbers” yet be to discovered!

I quickly spotted some triangles in the above group that do not fit with the formula. That is something that I had been looking for, because I have wondered if all integer right trangles fall into some predictive formula or another. Not having done the k(n) to them yet, I did make a quick spreadsheet and found that within the examples you have listed, none of the base integer right triangle's legs share any common prime factors. Prime numbers are another hobby of mine. This was the first time I had become aware of that particular nugget of information.

 

Thanks for the data Craig! I am going to try and automate some of this data so I can root out patterns... so much to do...

 

Bill

[Pyrotex]

Here is the reference to Katabataks & Conway's cellular automata:

Turtle,

I will try to upload an Excel spreadsheet with an implementation of a 2-dimensional ("ring") cellular automaton. It produces very pleasing patterns (Life Forms, or Actors) which can be seen to move around the ring as the generations pass.

Tell me if you like it.

Pyro aka Nelson

[Pyrotex]

... And as you speak of number patterns I am reminded of a pattern I have kept in my pocket for some 25 years. It is a collection of right triangles. Specifically, integer right triangles. I am curious about them, and am wondering if this cave might tell us more about their nature....Bill

Bill,

I am sorry this took so long to find. In this file, I called the right triangles, PIRTs, not PERTs. Prime Integer Right Triangles.

 

In this Excel spreadsheet, I list over 500 PIRTs, grouped by Delta (hypotenuse minus longest side). Then I plotted them, again by Delta.

 

Visually, we see some interesting things about PIRTs. Some, like Delta=1, appear to plot a curve of points that are nearly equidistant. But Delta=9 PIRTs plot in pairs; a pair of nearby points, a long gap, another pair of nearby points. Delta=25 PIRTs plot in groups of 4. Delta=98 PIRTs plot in groups of 6.

 

Take a look:

[Pyrotex]

Bill,

Another observation on PIRTs. Prime Integer Right Triangles.

 

I am sorry if you guys have already discovered this. I've been too busy to track down and read all the PIRT threads. But this is REALLY BIG.

 

What I call Delta, or somebody else called J, is the difference between hypotenuse and longest side. For a simple 3,4,5 PIRT, Delta=1. The question asked by someone (?) is what Deltas are allowed? Here's the answer.

 

Deltas fall in two SUPER CLASSES. I call one of them Odd Deltas, the other Even Deltas.

 

Odd Deltas = the squares of ODD integers.

1,9,25,49,81,121,169, etc.

 

Even Deltas = TWO times the square of ALL integers.

2,8,18,32,50,72,98,128,162,200, etc.

 

By inspection, you can see that there will never be any Delta that falls in both super classes.

 

I'm sure one of you has already seen this, perhaps, or maybe even proved that Delta (J) must be = (2n-1)^2 or 2(n^2) for all positive n>0.

[TheBigDog]

This is great stuff Pyro! I had not seen the "supersets" angle yet. I have been focusing on making a complete set of integer right triangles within a certain range, and am then going to try and play with ways to graph them so that patterns emerge. This spreadsheet and additional info really helps!

 

I am working on it, but you will see my progress slow a bit. I need to work on balancing my hobby time.:) But that only means more time with the kids.:cup:

 

Thanks!

 

Bill

[TheBigDog]

I have crudely fashioned a program that (I hope) gives me all the numbers I need for a complete analysis of right triangles. It has a simple user interface, and creates two text files as its output.

 

The first file is "Triangles.txt". It is comma separated values of each IRT found in a sequential search base upon side a and going up. Not efficient, but very thorough (I hope). The columns are as follows for the file...

 

a,b,c,k(a+b+c),k(a),k(:cup:,k©,k(c-:),k(b-a),k(c-a)

 

That covers the sides it pretty well (I hope).

 

The other file, "Factors.txt", is a list of the prime factors of all the sides found in the triangles search. It does not re-factor a number already in the file, so this will be joined to the triangles file when I play with this in Access. It has two columns as follows...

 

value, factor

 

I have attached the program is anyone want to play with it. It is written in vb.net, so if you are in Windows and have the .net framework installed it can just be detached and run. It will create the text files local to where the program is detached.

 

Next step is to begin searching the data graphically...

 

Bill

[TheBigDog]

___Hey Bill; how about pasting a portion of the output in the first text file here? This is not my machine & I can't download or use the program you mention. Great stuff you brought to the cave; thanks.

 

 

 

 

:singer:

OK, Turtle, I'll see what I can do...

 

Here are the first 100 PIRTs (Perfect Integer Right Triangles - Acronym courtesy of Pyrotex)

 

The legend is as follows...

a = short leg of triangle

b = long leg of triangle

c = hypotenuse of triangle

j = c-b

i = b-a

abc = sum of sides

m = c-a

Then follow the k(n) of those same values.

 

a,b,c,j,i,abc,m,kabc,ka,kb,kc,ki,kj,km
3,4,5,1,1,12,2,3,3,4,5,1,1,2
5,12,13,1,7,30,8,3,5,3,4,1,7,8
7,24,25,1,17,56,18,2,7,6,7,1,8,9
8,15,17,2,7,40,9,4,8,6,8,2,7,9
9,40,41,1,31,90,32,9,9,4,5,1,4,5
11,60,61,1,49,132,50,6,2,6,7,1,4,5
12,35,37,2,23,84,25,3,3,8,1,2,5,7
13,84,85,1,71,182,72,2,4,3,4,1,8,9
15,112,113,1,97,240,98,6,6,4,5,1,7,8
16,63,65,2,47,144,49,9,7,9,2,2,2,4
17,144,145,1,127,306,128,9,8,9,1,1,1,2
19,180,181,1,161,380,162,2,1,9,1,1,8,9
20,21,29,8,1,70,9,7,2,3,2,8,1,9
20,99,101,2,79,220,81,4,2,9,2,2,7,9
21,220,221,1,199,462,200,3,3,4,5,1,1,2
23,264,265,1,241,552,242,3,5,3,4,1,7,8
24,143,145,2,119,312,121,6,6,8,1,2,2,4
25,312,313,1,287,650,288,2,7,6,7,1,8,9
27,364,365,1,337,756,338,9,9,4,5,1,4,5
28,45,53,8,17,126,25,9,1,9,8,8,8,7
28,195,197,2,167,420,169,6,1,6,8,2,5,7
29,420,421,1,391,870,392,6,2,6,7,1,4,5
31,480,481,1,449,992,450,2,4,3,4,1,8,9
32,255,257,2,223,544,225,4,5,3,5,2,7,9
33,56,65,9,23,154,32,1,6,2,2,9,5,5
33,544,545,1,511,1122,512,6,6,4,5,1,7,8
35,612,613,1,577,1260,578,9,8,9,1,1,1,2
36,77,85,8,41,198,49,9,9,5,4,8,5,4
36,323,325,2,287,684,289,9,9,8,1,2,8,1
37,684,685,1,647,1406,648,2,1,9,1,1,8,9
39,80,89,9,41,208,50,1,3,8,8,9,5,5
39,760,761,1,721,1560,722,3,3,4,5,1,1,2
40,399,401,2,359,840,361,3,4,3,5,2,8,1
41,840,841,1,799,1722,800,3,5,3,4,1,7,8
43,924,925,1,881,1892,882,2,7,6,7,1,8,9
44,117,125,8,73,286,81,7,8,9,8,8,1,9
44,483,485,2,439,1012,441,4,8,6,8,2,7,9
45,1012,1013,1,967,2070,968,9,9,4,5,1,4,5
47,1104,1105,1,1057,2256,1058,6,2,6,7,1,4,5
48,55,73,18,7,176,25,5,3,1,1,9,7,7
48,575,577,2,527,1200,529,3,3,8,1,2,5,7
49,1200,1201,1,1151,2450,1152,2,4,3,4,1,8,9
51,140,149,9,89,340,98,7,6,5,5,9,8,8
51,1300,1301,1,1249,2652,1250,6,6,4,5,1,7,8
52,165,173,8,113,390,121,3,7,3,2,8,5,4
52,675,677,2,623,1404,625,9,7,9,2,2,2,4
53,1404,1405,1,1351,2862,1352,9,8,9,1,1,1,2
55,1512,1513,1,1457,3080,1458,2,1,9,1,1,8,9
56,783,785,2,727,1624,729,4,2,9,2,2,7,9
57,176,185,9,119,418,128,4,3,5,5,9,2,2
57,1624,1625,1,1567,3306,1568,3,3,4,5,1,1,2
59,1740,1741,1,1681,3540,1682,3,5,3,4,1,7,8
60,91,109,18,31,260,49,8,6,1,1,9,4,4
60,221,229,8,161,510,169,6,6,5,4,8,8,7
60,899,901,2,839,1860,841,6,6,8,1,2,2,4
61,1860,1861,1,1799,3782,1800,2,7,6,7,1,8,9
63,1984,1985,1,1921,4032,1922,9,9,4,5,1,4,5
64,1023,1025,2,959,2112,961,6,1,6,8,2,5,7
65,72,97,25,7,234,32,9,2,9,7,7,7,5
65,2112,2113,1,2047,4290,2048,6,2,6,7,1,4,5
67,2244,2245,1,2177,4556,2178,2,4,3,4,1,8,9
68,285,293,8,217,646,225,7,5,6,5,8,1,9
68,1155,1157,2,1087,2380,1089,4,5,3,5,2,7,9
69,260,269,9,191,598,200,4,6,8,8,9,2,2
69,2380,2381,1,2311,4830,2312,6,6,4,5,1,7,8
71,2520,2521,1,2449,5112,2450,9,8,9,1,1,1,2
72,1295,1297,2,1223,2664,1225,9,9,8,1,2,8,1
73,2664,2665,1,2591,5402,2592,2,1,9,1,1,8,9
75,308,317,9,233,700,242,7,3,2,2,9,8,8
75,2812,2813,1,2737,5700,2738,3,3,4,5,1,1,2
76,357,365,8,281,798,289,6,4,6,5,8,2,1
76,1443,1445,2,1367,2964,1369,3,4,3,5,2,8,1
77,2964,2965,1,2887,6006,2888,3,5,3,4,1,7,8
79,3120,3121,1,3041,6320,3042,2,7,6,7,1,8,9
80,1599,1601,2,1519,3280,1521,4,8,6,8,2,7,9
81,3280,3281,1,3199,6642,3200,9,9,4,5,1,4,5
83,3444,3445,1,3361,6972,3362,6,2,6,7,1,4,5
84,187,205,18,103,476,121,8,3,7,7,9,4,4
84,437,445,8,353,966,361,3,3,5,4,8,2,1
84,1763,1765,2,1679,3612,1681,3,3,8,1,2,5,7
85,132,157,25,47,374,72,5,4,6,4,7,2,9
85,3612,3613,1,3527,7310,3528,2,4,3,4,1,8,9
87,416,425,9,329,928,338,1,6,2,2,9,5,5
87,3784,3785,1,3697,7656,3698,6,6,4,5,1,7,8
88,105,137,32,17,330,49,6,7,6,2,5,8,4
88,1935,1937,2,1847,3960,1849,9,7,9,2,2,2,4
89,3960,3961,1,3871,8010,3872,9,8,9,1,1,1,2
91,4140,4141,1,4049,8372,4050,2,1,9,1,1,8,9
92,525,533,8,433,1150,441,7,2,3,2,8,1,9
92,2115,2117,2,2023,4324,2025,4,2,9,2,2,7,9
93,476,485,9,383,1054,392,1,3,8,8,9,5,5
93,4324,4325,1,4231,8742,4232,3,3,4,5,1,1,2
95,168,193,25,73,456,98,6,5,6,4,7,1,8
95,4512,4513,1,4417,9120,4418,3,5,3,4,1,7,8
96,247,265,18,151,608,169,5,6,4,4,9,7,7
96,2303,2305,2,2207,4704,2209,6,6,8,1,2,2,4
97,4704,4705,1,4607,9506,4608,2,7,6,7,1,8,9
99,4900,4901,1,4801,9900,4802,9,9,4,5,1,4,5
100,621,629,8,521,1350,529,9,1,9,8,8,8,7
100,2499,2501,2,2399,5100,2401,6,1,6,8,2,5,7

 

Enjoy!

 

Bill

[Pyrotex]

Bill,

...some interesting things about PIRTs. Some, like Delta=1, appear to plot a curve of points that are nearly equidistant. But Delta=9 PIRTs plot in pairs; a pair of nearby points, a long gap, another pair of nearby points....

I have been looking at the plot of PIRTS I uploaded. I made the Odd Delta plots have large, dark points, and the Even Delta plots have smaller, colored points, so I could distinguish the two super-classes. Each super-class is evenly distributed! But the distribution frequency is different between the super-classes. There are 7 Odd Delta curves in about the same domain as 9 Even Delta curves.

 

The curves have a pattern I call the Weirdness Factor. This is the tendency for most curves to consist of plots of points that group into "weird clusters". This needs some explaining. :confused: Some curves have an initial grouping cluster that is smaller than the repeating grouping that occurs after that. Here is what I can see from the plots:

 

Delta S-Class Weirdness, (initial weirdness)

1 Odd 0 No weirdness; points do not cluster

2 Even 0

8 Even 0

9 Odd 2,(2) Points cluster in groups of two, initial group = 2

18 Even 2,(2)

25 Odd 4,(1) Points cluster in groups of four, initial group = 1

32 Even 0

49 Odd 6,(2)

50 Even 4,(3) ??not sure

72 Even 2

81 Odd 2

98 Even 6,(5)

121 Odd 10,(3)

128 Even 0 ??not sure

162 Even 2,(1)

169 Odd 12,(3)

200 Even 4

 

So!!! Can anyone come up with a formula that defines at least the Weirdness of any specific Delta set of PIRTs, given the Delta? For example, W(81)=2.

Then, can we get a formula that defines the initial group of points? For example, W(169)=12 and w(169)=3.

 

Good luck to us all. :Waldo:

[TheBigDog]

Cool stuff Pyro! I began to play with the curve plots, and noticed the weirdness. I think there is a formula to be found for each curve that will include the weirdness.

 

What I have focused on is a different phenomena with the value i, where i = b-a.

 

All numbers in this column can only be factored by prime numbers that have appeared in this column. 1 is the exception and appears 5 times in the first 19271 PIRTs. It looks like there may be a formula to predict this curve of 1's too.

 

The values of i begin 1,7,17,23,31,41,47,49... 49 is the first non-prime value. And it, like all non-prime values in this list can only be factored by p rime values already appearing in this list. So none of the numbers are divisible by 2,3,5,11,13, as they never appear as a value of i. This is holding true for all PIRTs with a value of a up to 20000.

 

I thought this was facinating!

 

Bill

[Pyrotex]

...I have focused on is a different phenomena with the value i, where i = b-a ...The values of i begin 1,7,17,23,31,41,47,49...2,3,5,11,13, as they never appear as a value of i. This is holding true for all PIRTs with a value of a up to 20000.

I thought this was facinating!

You are correct! This IS fascinating!!! Good catch!!!

I'm assuming that i = b-a means the difference between the longest side and the shortest side.

So i ONLY has factors that are... what? Every 4th prime number, starting with 1???? If that's true, it would be beyond fascinating!!!!! :Waldo:

[TheBigDog]

Here is my Excel file with all PIRTs up to a=20000. It includes the k(n) for all values, auto filters are on for all columns and graph of a and b embedded to the data.

 

Oops! :Waldo: Too big to attach. If you would like a copy I can send via e-mail.

 

Bill

[TheBigDog]

Another pattern!

 

j = c-b is the same set of values (the two supersets) as m = c-a. Except that m cannot equal 1. I wonder if this holds the key to the weirdness?

 

Bill

[Pyrotex]

Another pattern!...

And yet another, maybe. did you notice in YOUR plot of the first 10,000 or so PIRTs, that the parabaloid curves of the PIRTs themselves, also have orthogonal arcs connecting them? Every so often a member of each PIRT lines up somewhat horizontally to form the "stringers" of a WEB.

 

Is this significant? What is its meaning?

Where is Mister Universe when you need him??

[Pyrotex]

...I wonder if this holds the key to the weirdness?...

I have a friend here at work--PhD in math--does quaternions in his head just for fun. I mentioned our PIRT project and showed him my chart. He thinks we should publish. No kidding.

 

Can you see US with an article in some recognized math journal? With our (real) names on it? Tooooooooo freakin kool, dudes! :Waldo:

 

Anyone know how to set up an academic journal search in Google? Probably should look for journals that print articles on "Diophantine Math" which is the study of equations with integer solutions. Sounds like us.

[TheBigDog]

And yet another, maybe. did you notice in YOUR plot of the first 10,000 or so PIRTs, that the parabaloid curves of the PIRTs themselves, also have orthogonal arcs connecting them? Every so often a member of each PIRT lines up somewhat horizontally to form the "stringers" of a WEB.

 

Is this significant? What is its meaning?

Where is Mister Universe when you need him??

Awsome catch Pyro!

 

There are curves in 4 distinct directions. It is like graph paper with diagonal lines overlaying the regular lines. And it is on a warped surface. Every dot is an intersecion of 2 or more lines. 'j' isolates each horizontal line. I need to find a way to hi-lite or isololate the other curves on the graph...

 

To see this place the cursor over the point at 696,697 and follow it in 4 directions. Like one of those hidden 3D pictures it jumps right out at you.

 

This is great stuff! :Waldo:

 

Bill

[TheBigDog]

I have a friend here at work--PhD in math--does quaternions in his head just for fun. I mentioned our PIRT project and showed him my chart. He thinks we should publish. No kidding.

 

Can you see US with an article in some recognized math journal? With our (real) names on it? Tooooooooo freakin kool, dudes! :Waldo:

 

Anyone know how to set up an academic journal search in Google? Probably should look for journals that print articles on "Diophantine Math" which is the study of equations with integer solutions. Sounds like us.

That would be so awesome! I waited an hour after reading this to respond so my ego could settle at its new equilibrium. I am going to keep plugging away at isolating those curves by making an interactive method for doing analysis of the dataset. But real life has me busy today, so it will have to wait awhile.

 

Bill