Katabatak Powers

[Turtle]

The Graphs
(click for full-size image)

mod one remainders of powers


mod two remainders of powers


mod three remainders of powers


mod four remainders of powers


mod five remainders of powers


mod six remainders of powers


mod seven remainders of powers


mod eight remainders of powers


mod nine remainders of powers


mod ten remainders of powers


mod eleven remainders of powers


mod twelve remainders of powers


mod thirteen remainders of powers


mod fourteen remainders of powers


mod fifteen remainders of powers


mod sixteen remainders of powers


mod seventeen remainders of powers


and so on...

[Turtle]

The original Katabataks thread is now unwieldy & fragmented. No worries though! Please; put down that sword! I have some digging to do, for graphs & lists in my archive, so it's a shovel for me.

In the meantime, the bar's open & the smoking lamp is lit.

[belovelife]

i am unfamiliar with these
could you define please

[Turtle]

Originally Posted by belovelife

i am unfamiliar with these
could you define please

Oh I please alright. While the only published Katabataks explication is in the thread I linked to in post #2, starting from scratch for the Powers here is just what we're about. Let's start with a definition of Powers:
Exponentiation - Wikipedia, the free encyclopedia

Originally Posted by Wiki

...The exponentiation a^n can be read as: a raised to the n-th power, a raised to the power [of] n or possibly a raised to the exponent [of] n, or more briefly: a to the n-th power or a to the power [of] n, or even more briefly: a to the n. Some exponents have their own pronunciation: for example, a^2 is usually read as a squared and a^3 as a cubed. ...

Now your part. Please make a notepad list of at least the first twenty Powers-of-Two, i.e. the Perfect Squares.

[Turtle]

The convention I have used in my graphs is to list these sets of powers horizontally & left-to-right. Like this list of Powers-of-Two. (The consecutive positive integers above the list give both the value to be raised to the power, as well as the ordination of an element in the list. Which is to say, the fifth power of two is five to the power of two. How convenient! )



Another convention I have is to pause & look over a result. I spy with my little eye that the ending digits of the Perfect Squares (Powers-of-Two) have a repeating ten-digit-long pattern:
{1 4 9 6 5 6 9 4 1 0 }
I also spy that the pattern does not use/contain the digits 2, 3, 7, or 8.

That's another break.

[Turtle]

So many numbers, so little time! Those of you already familiar with Katabataks, if you have seen yet, as I have, that I have done volumes of uneccessary calculations over many years , please bear with me.

So we have above our base ten list of the ordered set of powers of two, and an infinite pattern of repeating end digits from it. Pause here to consider what we commonly take from the last digit of a base ten number. We know divisibility by two (odd/even), and we know divisibility by five & ten. We also know, but perhaps don't think of in these terms, that the last digit of a base ten number is the remainder/residue of the number after dividing by ten.

A last note before giving a list of powers of two in base eleven, is to say that nothing about the graphs is changed by my realization; only the work involved to derive them.

powers of two: base eleven

[Don Blazys]

Interesting. Now we have a palindromic pattern of last digits that is eleven digits long and does not contain 2, 6, 7 or 8.

Don.

[Turtle]

Originally Posted by Don Blazys

Interesting. Now we have a palindromic pattern of last digits that is eleven digits long and does not contain 2, 6, 7 or 8.

Don.

So it is. Well, the terminating 0 in the pattern makes the whole non-palindromic, but the the ten-element section has the goods.

Now from here it is apparent we might try this with all bases (or as many as we can carry), and as luck has it that is what I have done & recorded with my graphs.

Now I have to go aside because I was getting to the patterns of remainders by applying the Katabatak function (my name) to the powers written in different bases. This function has other names (can't find a Wolfram article just now) but it is simply the re-iterative adding of a numbers digits until arriving at a single digit.

So, taking the base eleven powers of two one at a time and operating on them with the K function gives us this:

Now what we have is the pattern we found in the ending digits of the base ten list of powers of two, namely {1 4 9 6 5 6 9 4 1 A}. Note the last digit/symbol differs in that the base ten ending-digit list/pattern terminates with 0 and the base eleven Katabatak transform pattern gives A, but since A is ten and ends in 0 in base ten then it is really the same result, i.e. a remainder of ten.

So I never needed to add the digits after changing base, rather only needed to take the last digits from the new base. Well, as I came at it from the adding digits algorithm in the first place I don't feel as bad as I could over so many years of extra work.

More to come - as if - and I am continuing to add graphs to the first post.

[Turtle]

Mmmmmm... tastes like peppermint candy! I do recall noticing this ending-digit vs. K-function business and discussing it some with Craig in the Katabatak thread, but I also recall he wasn't sure what I was talking about & we moved on.

Well, we're hare now so on with the race. Now I was just off thinking that with odd-base numeration we can't tell even/odd from just the last digit. That's odd in and of itself.

OK. Back to 'splainin' the graphs. Iteration, iteration, iteration. Infinite pattern repetitions of the infinite squares written in infinite bases, yada, yada, yada, bleh, bleh, bleh. Wash, rinse, repeat. We're headed back to do it all again for the first time with the cubes, the powers of three, the set n^3 for n=1,2,3... . Gotta go rummage in the archive.....................

[Turtle]

Got it. Rather than edit out the powers of three separately, this is the the full Monty of remainders of powers mod nine in base ten through powers of ten. Fortunately, the way I wrote them out gives us the ending digits as well. I get some idea also from the way I wrote out the lists, of how it was I never saw the ending-digit pattern matching the K-pattern of next lower base; I never wrote them together.
This page does however illustrate an emergent property of remainders of powers that caught me by surprise, and we will get to that once the prelims are out of the way.