maconvert
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between 1 billion and 42 trillion or between 1 trillion ...? You mention both
And after thinking about it - no way!
Even if you choose the smallest set of integers 2,3,4,5 you have maximum values of their exponents as 45,28,22,19 respectively - otherwise you go over 42e12 with just one of the terms when the others are zero.
ie 2^(0 up to 45) + 3^(0 up to 28) etc
There is therefore an absolute max (and in fact it will be a lot smaller) of combos (45*28*22*19) - its just over 500k ignoring the fact these max exponent values are based on the others being zero.
So it can only cover a tiny fraction of the numbers betwee 1e12 and 42e12
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you could get arbitrarily close if you allowed non-integer exponents - but what would be the point?
The maximum exponent value would be 35 (36 total values including 0).
I basically want to create an algorithm that runs through every combination of A^a + B^b + C^c + D^d (1,679,616 total possible equations) and compares each sum to the number I want to represent.
When it finds a match, it generates the 4 character code (numbers and/or letters) and presents that to the user.
I know that this is a relatively inefficient brute force method and I'm aware that a huge proportion of the results will fall outside of the range that I specified, but that's OK. The algorithm will quit when (if) it finds a match, but even if it takes 2 hours to arrive at a result, that's fine. Speed is not an issue in this case.
BTW, to your point regarding the maximums, I guess if I use A = 2, B = 3, C = 5, and D = 7, then I should limit my exponents to 35, 28, 19, and 16 respectively.
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Hello,
I'm working on a computer problem, but, before I waste too much time going down this road, I thought I should pick the brains of you math geniuses.
Basically, I want to be able to represent any number between 1 billion and 42 trillion using just 4 characters (numbers and/or letters).
This is what I've come up with so far...
A^a + B^b + C^c + D^d = LARGE NUMBER
a, b, c, & d are exponent variables that can range from 0 to 35 (0,1,2,...X,Y,Z)
A, B, C, & D are bases that are integer constants that never change.
Basically, if someone is given the 4 exponent values and the bases are already known, they can extrapolate the number value from that.
For example, if I decided to always use A = 2, B = 3, C = 5, and D = 7, the exponent sequence Z9E2 would correspond to 2^26 + 3^9 + 5^14 + 7^2 = 6,170,644,221.
So, what I want to know is this:
Are there particular values of A, B, C, and D that, depending on the values of a, b, c, and d, can generate every single number between 1 billion and 42 trillion?
Is there NO set of 4 base values that can generate every single number between 1 billion and 42 trillion?
Will any values of A, B, C, and D work?
I look forward to your replies.
Thanks in advance!
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Representing Large Numbers with Fewer Characters
in Linear Algebra and Group Theory
Posted · Edited by maconvert
Now, that I've eliminated exponents that are over the maximum, the number of equations is 354,960.
The bottom line, and this is not a deal breaker, is that the number I want to encode into a 4 character string, must be the sum of one of these 354,960 equations.