Need an Algorithm for this Integer problem. Can you figure it out? 
Looking for an algorithm to perform the following mathematical function:

Given an INTEGER up to a maximum of 21 digits, decompose the number
into a formula like this:

X^y + W^v + S^t = Number 
(X^Y means X to the power of Y and so forth)

where

the bases X, W, S are integers between [2-255]
and the exponents y,v,t are also integers between [0-255].

Lets see if there is a math genius that can figure this out with an
algorithm that will work for all numbers up to 21 digits

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Request for Question Clarification by eiffel-ga on 19 May 2006 09:49 PDT

Hi zeusdman-ga,

Would you accept, as an alternative answer, a straightforward proof
that this is impossible?

Regards,
eiffel-ga

You can't get 1 or 2 for starters...

Why the constraint of 2-255 on the bases?

Yup - impossible.  You want an algorithm to decompose 10^21 different
values, but your formula (with constrants) cannot create 10^21
different values.  (It can create values much larger than 10^21, but
you need your formula to be able to represent ALL different integers
up to 10^21.)

Good point oblok.  Total number of ways to choose X,W,S,y,v,t is
254^3*256^3 = 2.75E14, more than a million times too small.

zeusdman,

When you write "formulas _like_ this," do you mean to allow a variable
number of terms on the left and do you mean to allow repetitions of an
exponent, i.e.,

3^0=1
3^0 + 2^0=2

but not 3^0 + 3^0 + .... = 1000?

(At the moment, I don't know if this will allow a solution; I'm just
asking if this interpretation is a possibility.)

@zeusdman-ga 
 
I can see what you are getting at.

A 6 byte unsigned integer looks like :-

  Debug.Print Format(255# ^ 6, "#########################")
  274941996890625  eg: 15 digits which is a lot less than 21 digits

By making the first two bytes into X^Y  you can get some incredibly
large numbers, but to get a complete range of numbers you need to be
able to fill in the 'holes'

With two bytes the maximum range that you can 100% describe is 65025

255^1 255
255^2 65025
255^3 16581375
255^4 4228250625

Notice how the jumps become unfillable

My guess is that you have a legacy system with a 6 byte Float field
(Pascal) and you want to expand its range.

Unfortunately pure binary is the only way of storing a precise number,
IEEE or BCD will give you a (wildly inaccurate) approximation.