To calculate All powers (exponents) of an integer from a single long division:
^ means raise to a power
precision means how many digits to allow for each power
Insufficient precision causes carry over mixing of powers.
Simple example: Precision=3 digits, the powers of 2=:
1/(1000-2) = 0.001 002 004 008 016 032 064 128 256 513_024
See the powers of 2 to 3 digit precision until the next power
gets too large to fit. Are these rational numbers? Yes!
The calculation provides quotients between 0 and 1, with a predictable pattern such that the powers use equal amounts of digits selected by the precision, which is the power of 10 used. The long division is a fraction, using this form:
Power series = _______________
The problem can be simplified using only integers and a very long division that assumes plenty of zeroes after the 1 without a decimal point. Then the powers appear one digit at a time in the elementary (chalkboard division) in simple convenient blocks of digits grouped according to the precision term, which is the number of digits you need for the powers you want.
N means the number whose powers you want. I wrote it that way instead of ((10^p)-n))100000000000000000...00
because I used parenthesis for the long division symbol also.
This is a simple minimal algorithm for calculating exponents that is part of an algorithm for generally larger, faster number theory experiments for recreational mathematics purposes. The long division can be considered a spigot function for powers, which provides one digit at a time. It is very efficient since large powers require only the memory needed to hold them if it is programmed for the use of determining a single power, since the lower powers are not going to be used, and when the desired power is reached the whole long division may terminate especially since it will begin overflowing its preselected precision.
Some people have doubted this works in other bases due to it somehow failing in binary. I am confident that if it doesn't work in binary, it will work in hexadecimal to give the desired result. It would be a surprise to me that it might not work in binary since it reminded someone else of a function for the powers of e.
Why this works: Normally powers are done by multiplying a number by itself over and over again. I have not proven that using division is more efficient but only that it seems to be, since more information flows from a single fundamental arithmetic function. An interesting question suddenly emerged as to whether this result can happen without the fraction, that is, with the reciprocal of the fraction, and in the interest of faster math, I am looking into that. My math system is not floating point because I need arbitrary precision. My own number theory is generally based only on integer means and results. A division is a fraction of two integers, resulting in a quotient and maybe a remainder, which represents the fraction of integers on the right side of the decimal point. Even my theory of infinite data from Champernowne's Constant is really a theory of infinitely fast counting of integers. (Proper use of the word theory is the how, not the maybe. A maybe is called a hypothesis. An example of this mistake is the uses of the term Conspiracy Theory.
Division has many other unexpected uses, for example:
Pascal Triangle Integer,
Approximating Irrational Numbers,
The above nodes have been deleted by moderators for various deficiencies and may be reworked.