^{10}+3

^{30}.

In other words if we didnt know 205891132095673 is equal to 2

^{10}+3

^{30}how would we find it, is it even possible?

What about other numbers, prime and non-prime.

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In other words if we didnt know 205891132095673 is equal to 2

What about other numbers, prime and non-prime.

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There used to be a site called Plouffe's Inverter that did this sort of thing (for approximate real numbers -- probably not for huge integers) by essentially just looking up your value in a database, but neither that nor its successors seem to be working. In searching for information about that, I learned that Wolfram Alpha can do something similar, as in https://www.wolframalpha.com/input/?i=closed+form+1.4142 .

But it doesn't work for your number.

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Now, you started by asking

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Many people have an exaggerated sense of what math can do, in part because we tend to teach only the problems for which we have methods to teach. Math can't do everything you can imagine! In fact, it is easy to make up a problem that is unsolvable (in principle, not just in the sense that I don't know how).That is what i already had in mind altho i find it strange with all the advanced math we have we cant extrapolate those values.

But what does "extrapolate those values" mean? You're not talking about taking a set of values and finding a likely future value based on the trend, which is what "extrapolate" means. Your question is about finding one of the infinite variety of possible expressions that yields a single given number.

No routine method can guarantee an answer to that question (in a finite amount of time). Yes, if you define what operations you are allowing in your expressions and make an ordered list of all expressions using any numbers together with those operations, in any order, you could hope to eventually find it; but that's a computer program, not advanced math, and would be horribly inefficient. (Your particular expression might be found soon enough; but what if the only way to get a particular number was as the difference of two huge powers of huge numbers?)

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Gauss reportedly said something similar when asked about Fermat's Last Theorem, I believe. It led me to think that perhaps he had some prescience of Goedel's Incompleteness Theorem....In fact, it is easy to make up a problem that is unsolvable (in principle, not just in the sense that I don't know how)...

Maybe but i'd rather say it's not the math that's limited but our understanding of it. We must think creatively, flexibly, beyond limiting patterns.Many people have an exaggerated sense of what math can do..

I understand we don't know of a way to directly reverse the number to a simple equation, as i suspected altho i hoped for the opposite.

Like i said i thought of index with many simple equations and all possible interactions of them but index alone is not enough 'cause there is simply too many possible numbers to address them all.

Goal is not to reduce any number, but only up to certain number of digits, let say 15 like in my example.

If any 15 digit number can be reduced to an expression with aprox half number of digits, you do realize multistage lossless near infinite compression of data becomes simple task, like Sloot encoding which compressed a whole HD movie down to 4 kilobytes.

If you're not familiar with it, he had a library of 370Mb, about 70Mb for each kind of data, movies, sound... He was obviously comparing data to data in his library, generating keys of minimal size which could regenerate the original on the remote computer from the same library.

Only way this can be done even in theory is if he reduced big chunks of numbers to small mathematical expressions. Exactly how he did it is not clear.

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Do you even know what does an expression mean in this context?Only thing that is incredibly stupid is your comment. There are not ''infinite number of expressions that equal a given number'', unless you count expressions longer than the original number which is idiotic.

(100 - 2) is an expression. Which is equal to 98. There are infinite number of expressions - e.g

49 * 2 or

96 + 2 or

7

..

ad infinitum

That will equal to 98 (for example).

You said - "...unless you count expressions longer than..."

How do you measure the length of an expression?

Please knock off that chip off of your shoulder .....

Do YOU know what an expression means in THIS contex. You took an expression weighting 4 bytes or more to express a number worth 2 bytes, pure idiocy. Whole point is to REDUCE the amount of information that needs to be sent.Do...

So you knock off that chip off of your shoulder and stick it in your head, at least it won't be vacuum any more.

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Claiming that an expression longer than the number it represents is not an expression, however, is silly. We need to define our terms before we use them in an argument.

I don't want to use the word "stupid", which does not produce useful communications, but we do need to ask the OP to face reality. Think about what is being

Suppose that an "expression", whatever definition we give for that, consists of some combination of 10 digits, 10 operations, 26 letters (used for function names, say). Then 7 such symbols can form (10+10+26)^7 = 435,817,657,216 distinct strings; that's about 4*10^11, which is far less than 10^15, so we can't represent every number. And I've been very generous here in allowing any word at all as a function. Consider that many of these strings will be meaningless (not expressions at all), and many will produce duplicate numbers. The claim is nonsense.

By the way, I finally looked up Sloot to see what the background of the question is, and everything fits together. This is a fantasy question, and we should drop it.

''Claiming that an expression longer than the number it represents is not an expression, however, is silly'' - I never said it is not an expression, just not a useful one.post #8, which shows a little more of what the OP has in mind. The question is not really about finding "the" expression that yields a given number, as was stated, but about associating each number up to, say, 10^15, with some expression for the number, which he expects to be less than 8 symbols (half as many), in order to arrive atlossless compression. If that had been stated in the original question, some of what has been said here would not have been said. The OP's context was misrepresented, so we are talking past one another.

Claiming that an expression longer than the number it represents is not an expression, however, is silly. We need to define our terms before we use them in an argument.

I don't want to use the word "stupid", which does not produce useful communications, but we do need to ask the OP to face reality. Think about what is beingassumedhere without any attempt at proof. Why should we think that every number can be calculated by an expression with half as many symbols as the number itself? This turns out to be self-refuting.

Suppose that an "expression", whatever definition we give for that, consists of some combination of 10 digits, 10 operations, 26 letters (used for function names, say). Then 7 such symbols can form (10+10+26)^7 = 435,817,657,216 distinct strings; that's about 4*10^11, which is far less than 10^15, so we can't represent every number. And I've been very generous here in allowing any word at all as a function. Consider that many of these strings will be meaningless (not expressions at all), and many will produce duplicate numbers. The claim is nonsense.

By the way, I finally looked up Sloot to see what the background of the question is, and everything fits together. This is a fantasy question, and we should drop it.

As for the rest of your post i figured myself it is futile to try to force a number to an expression half its length which is also composed of NUMBERS thus having final value which never allows for all possible variations of the bigger number. It was just one idea, an attempt of figuring out how Sloot did it.

I don't want to use the word stupid either but thinking in limited, fixed terms would NEVER allow Sloot to achieve what he achieved.

Just one little addition, you say 26 letters, yet there are 255 extended ASCII codes, each taking one byte. So 255 - 20 codes for numbers and 10 operations there are 235 left for functions, which results in one order of magnitude more numbers, but still far less than all possible 15 digit numbers.Suppose that an "expression", whatever definition we give for that, consists of some combination of 10 digits, 10 operations, 26 letters (used for function names, say). Then 7 such symbols can form (10+10+26)^7 = 435,817,657,216 distinct strings; that's about 4*10^11, which is far less than 10^15, so we can't represent every number.

This is the original post. It says nothing about bytes, does not stipulate anything about "length," let alone define what "length" means in this context, does not explain how many symbols are permitted nor their meaning,, and shows no indication whether the equivalent expression is to be unique. It is beyond stupid.^{10}+3^{30}.

In other words if we didnt know 205891132095673 is equal to 2^{10}+3^{30}how would we find it, is it even possible?

What about other numbers, prime and non-prime.