Katabatak Powers

[Don Blazys]

The even exponents give us eight digit palindromic patterns while the odd exponents do not. That's "odd".

The exponents (2 and 8), (3 and 9), and (4 and 10) result in the exact same patterns. That's "even odder". (I know... "even odder" is bad grammar... but I like the "play on words" and the way it sounds! Try saying it really fast four or five times in a row!)

[Turtle]

Originally Posted by Don Blazys

The even exponents give us eight digit palindromic patterns while the odd exponents do not. That's "odd".

The exponents (2 and 8), (3 and 9), and (4 and 10) result in the exact same patterns. That's "even odder".

Nicely seen. Now what my cryptic algebraic terms along the left margin in the following graph represent is just that repetition of patterns across powers in base ten (mod nine remainders) which extends to infinity (moving vertically down the graph). In light of the recent ending digit realization I will start titling the graphs 'powers mod n remainders', but the graph associated with the list is titled as 'base ten' and only subtitled mod nine.

So anyway, the repeating length-nine mod nine remainder patterns(moving horizontally across the graph) found for the first 6 powers (n^2 -n^8) repeat vertically as a group across all powers in the same order starting with n^9 - n^15, then again n^16 - n^22, ad infinitum. The numbered colored boxes in the 6x9 upper-left rectangle of the graph is the complete pattern, and the other graph elements are separations of the individual groups of like remainders making up the complete graph. Colors are keyed to remainder values to emphasize group patterns within the main pattern. Thanks for playing.

(click for full-size view)

[Turtle]

To cut down on my confusion, I'm adding headings above the graphs in post #1 that give just the divisor used to derive the remainders. Once we have a grip on how the graphs are constructed, then it's off to just working with the graphs.

In the mean time & before moving on, I have kept worrying the business of telling odd/even-ness of a number written in an odd base. I worried it right down to the conclusion that you have to use all the digits. How inconvenient!!

[Turtle]

While I put the finishing touches on three new graphs, I thought I'd point out some peculiarities already developed in the dozen graphs put up so far. Any and all are open for discussion.

1) notice some graphs have n^1 included and some not. this is because it does not recur in the repeating part of some patterns and so I left it off.

2) notice that while the horizontal repeating component is always the same length as the divisor of the modulo function, there is no apparent rhyme or reason to the length of the vertical repeating component across higher powers. craig has done some calculations out to mod 500 and found no pattern, and i'll get that reference from the other thread presently.

3) notice the remainder eight pattern has a sub-pattern for n^2 that does not repeat again. the graph reflects my surprise and uncertainty about how to graph it. i was somewhat less surprised then tonight when calculating the mod 16 remainders to find that both n^2 and n^3 do not repeat, but that's a few days off.

that's a rap...erhm...wrap.

[Turtle]

Originally Posted by Turtle

...2) notice that while the horizontal repeating component is always the same length as the divisor of the modulo function, there is no apparent rhyme or reason to the length of the vertical repeating component across higher powers. craig has done some calculations out to mod 500 and found no pattern, and i'll get that reference from the other thread presently.
...

A couple new graphs done and added to post #1and here is that calculation by Craig. The left column is the length of the repeating (vertical) part, and to the right is the list of bases having that length.

Originally Posted by CraigD

I don’t find this strangeness. My arithmetic shows 17^2 mod 28 = 17^8 mod 28 = 17^(6*X+2) mode 28 = 9, and 17^3 mod 28 = 17^(6*x+3) = 13 for all the nonnegative values of X I tried.

My quick & dirty checking M code:
s e=3,n=17 f e=e:6 s a=n x "f i=2:1:e s a=a*n#28" w " ",a

I'll try de-mystifying this fundamentally – that is, account for it as a property of sets of modular numbers under exponentiation. My playtime will be in short supply ‘til next weekend, though, so this may be a while in coming.

PS: I recoded to avoid the limitation that was limiting me to bases up to 88, and produced the “pattern modularity to base” map up to about base 500, where my program became impractically slow.

Some of the missing modularities appeared in higher bases, and modularity 8 is revealed to no longer be unique to base 33. I suspect that continuing to look at higher bases would banish all of the missing and unique modularities (except 1).

Here it is:
1: 3
2: 4 5 7 9 13 25
4: 6 11 16 17 21 31 41 49 61 81 121 241
6: 8 10 15 19 22 29 37 43 57 64 73 85 127 169 253 505
8: 33 97 161 481
10: 12 23 34 45 67 89 133 265
12: 14 27 36 40 46 53 66 71 79 91 92 105 106 113 118 131 141 145 157 181 183 196 209 211 235 261 274 281 313 316 337 361 365 391 421 456 469
16: 18 35 52 65 69 86 103 137 171 193 205 256 273 321 341 409
18: 20 28 39 55 58 77 109 115 134 153 172 190 217 229 267 343 379 400 457
20: 26 51 56 76 101 111 151 166 177 201 221 276 301 331 401 441
22: 24 47 70 93 139 185 277
24: 225 289 417
28: 30 59 88 117 146 175 233 291 349 436 465
30: 32 63 78 94 100 125 155 187 199 218 232 249 280 309 342 373 397 435 463
32: 129 385
36: 38 75 96 112 136 149 186 191 223 248 260 271 286 297 305 334 352 371 381 433 445 482 495
40: 42 83 124 165 206 247 329 353 411 452 493
42: 44 50 87 99 130 148 173 197 259 295 302 345 388 393 442
44: 116 231 346 369 461
46: 48 95 142 189 283 377
48: 120 154 222 239 307 358 443 449 477
52: 54 107 160 213 266 319 425
54: 82 163 325
58: 60 119 178 237 355 473
60: 62 123 144 156 176 184 226 245 287 306 311 326 351 367 386 404 428 430 451 466 489 496 497
64: 257
66: 68 135 162 202 208 269 323 403 415 470 484
70: 72 143 214 285 427
72: 74 147 220 293 366 439
78: 80 159 238 317 475
80: 188 375 426
82: 84 167 250 333 499
84: 204 216 246 262 378 407 431 491
88: 90 179 268 357 446
90: 210 298 419
92: 236 471
96: 98 195 292 389 486

[Turtle]

While constructing the mod seventeen remainders pattern (still a day or two to go ), it struck me that the patterns having prime divisors have a 'full' pattern that is almost square, while composite divisors give the narrower patterns.

Reviewing the patterns posted so far in post #1, I see 'full' patterns for mod 3, 5, 7, 11, and 13.

[Turtle]

Originally Posted by Turtle

... it struck me that the patterns having prime divisors have a 'full' pattern that is almost square, while composite divisors give the narrower patterns.

Reviewing the patterns posted so far in post #1, I see 'full' patterns for mod 3, 5, 7, 11, and 13.

Mmmm....these are the same patterns that include powers of one (n^1) I mentioned earlier, and they are also characterized by the last power/row in the pattern having all remainders of one except the last column. This character appears in the graphs as a row of black squares at the pattern bottom.

[Turtle]

Originally Posted by Turtle

... it struck me that the patterns having prime divisors have a 'full' pattern that is almost square, while composite divisors give the narrower patterns.

Reviewing the patterns posted so far in post #1, I see 'full' patterns for mod 3, 5, 7, 11, and 13.

Hold on!! something skwewey here. Reviewing Craig's list (today that is funny. ) I find it agrees with my graphs to base seventeen (mod sixteen remainders), but it only has patterns to length 96 while going to base 500, so where are the other primes then? By my earlier assertion there ought to be a length 100 pattern for base 102, i.e. remainders mod 101. I can't find a base 102 entry anywhere on the list? Craig? Turtle down!

[Turtle]

i'll stick with my assertion and suggest craig's list is incomplete.
phrased informally, when taking the remainders of powers, if the divisor used for the mod function is a prime, the vertical pattern length (from my graphs) will be one less than the prime divisor and the remainders for integers raised to the power of one less than the prime will be 1, excepting for every prime'th remainder which will be 0.

also informally, the graphs prove special cases of fermat's last theorem. for example from the remainders mod nine graph, the pattern of cubes and every 6th power after is 1 8 9 1 8 9 1 8 9. now for x^n+y^n=z^n if we let n=3 then let x=1 & y=1 (remainders of every 3rd cube)) the sum is 2 and since 2 is never a remainder of dividing a cube by 9 then there is no solution.

that's a wrap.