MHB How to find functions & inputs whose output is a specific number

AI Thread Summary
The discussion revolves around finding a function and input pair that outputs a specific large number, n, while keeping the inputs relatively small. This problem is linked to data compression, where representing a bit string as a function can reduce its size. One proposed method involves expressing the number as sums or differences of large prime powers, but the effectiveness of this approach is questioned. An alternative suggestion is to define a simple function that adjusts the input to reach the desired output, highlighting the existence of infinitely many such functions. Overall, the conversation seeks efficient ways to express large numbers compactly through mathematical functions.
DaviFN
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I'm interested in the following problem: given a random number n (n can be gigantic), how do we find a pair function+input(s) whose output is n such that the input(s) are relatively small in size?

This problems arises in data compression; consider the bits that make up a file (or a substring of bits of the file) and treat it as a number (i.e. the bits are the binary representation of this number). If we could write a pair function+input(s) whose output happens to be the substring, this whole substring can be replaced by the function+input(s).

I've thought of expressing the number as sums (or differences) of relative big powers of prime numbers. Is this a good approach? And, if not, what would be a good one? And how to proceed?

Motivation of the question: A simples function like raising the nth prime number to a power S can result (depending on the values of p and S) on various outputs, each of which is unique (given that any number has only one prime factorization). If we pick p = 17 and S = 89435, for example, that's computationally trivial to compute (takes logarithmic time), and will result in a somewhat gigantic number. We can then generate a file whose bits are the same of the binary representation of this number (or at least some of the bits are). (This is just a rough example). The problem is going the other way: Given a bit string (hence, a number), how to express this specific bitstring with less bits (very few, actually) through a function that results in the number.

Any ideas/answers/comments are welcome!

This question is cross-posted https://math.stackexchange.com/questions/3275106/how-to-find-functions-inputs-whose-output-is-a-specific-number.
 
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