Δευτέρα 17 Σεπτεμβρίου 2012

The Powers of 2 and Why They Are Divisible by 2

[of course that's the provocative title which tries to obscure the answer that it gives
it's not a one thing it has two parts and the second tries to obscure the first
so a subdivision into two is necessary to discover the truth

subdivision into two ad infinitum?

of course the other way is to meld two into one
and they better be real not fony ones

ideally complimentary

here we have the evil intent to dissimulate by combining a straight statement with an evil dissimulatory one i guess that what the church calls the devil

you try to glue together two broken pieces of the same original one

not combine two pretentiously constructed elements

good vs evil

god and devil

or at least one is pretentiously constructed

and the melding of the two leads NOWHERE]

i wonder if that's what the salesmen mean when they say this beat up old car is brand new


ah the power of dissimulation

to make it appear

the appearance is the appearance

everybody gets the same perception / appearance (we will assume the retina is the same for all for now)

but how and where it is processed in the brain changes
]

i was thinking how to find a number that gives the longest possible division into 2 without being of course a power of two

i'm sure there is a theory somewhere developed or about to be developed or

we don't want the high brainy math
we want to keep it to the ground. you get me?


you got to start with an even number
because if you start with an odd you lost right off the bat

all powers of 2 have the pattern 2 4 8 6 2 4 8 6

in their rightmost digits

but then what?

what comes next to their left viccinity?

has it to be an even or an odd?

2 divided by 2 gives 1 so that loses right there if it also has an even to the left (which will be divided even to some number and leave no repercussions left for its rightmost friend)

so then you need an odd left to 2

such as 12 32 52 92: all divide to an even number so we still have hope for further subdivisions

what of the 4 he needs an even to its left otherwise he will fall into the odd hole when divided by two:

e.g. 14 -> 7

while 24 -> 12

and so on

and it doesn't matter what next the subdivision is guaranteed

112 -> 56

212 -> 106

now if you want two further subdivisions we got to choose a partner

an odd gives better subdivisions 112 56 28 14 7
212 106 53

and what if you chose 3 in the previous round to pair your 2

32

132 66 33
232 116 58 29

so that obviously has to do a lot

i'll stop right there since my brain is burning

and i'll go to a secluded place where there is no turning back
unless somebody hits me with a hammer in the head

and 2112 gives you a lot more repetitions subdivisions than 1112

obviously that has to do with the peculiarities of the numbering system

if you got a system with 3 digits 1 2 3 (ok and the 0)
2 would be 2 3 3
but then 4 is
10

oh jee that throws off my whole sceptics
but for the number 10 numbering system you get what you get


the other approach would be to ask how near to a perfect power of 2 it would have to be or how far

112 is closer to 128 than 212 (and 212 is closer to 256 then)
but 114 is closer and 114 -> 57 and that's it you fall into a hole

where in the landscape of numbers (whole numbers that is) would they be located?

it appears there are some "sweet" spots

let's try and graph these



and here for some higher intensity



(these are not normalized; i don't care for that)


we see the sweet spots are periodically populated  (the black is the abyss of the odd numbers)

see the dump see the vector of occurrences

c [1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1,
2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1,
3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2,
1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 3, 1, 2, 1, 4, 1,
2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3,
1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1,
5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2,
1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 9, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1,
2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4,
1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3,
1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 8,
1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1,
2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2,
1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1,
5, 1, 2, 1,  ]

you see the 9 in the middle for 512


that could be described by a recursive

it has the self similar of the fractalness

it abhorrs me to see these nice slithery pictures

we are creating a shiny surface here
and we leave the 2's and the 1's to live their lifes


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