What is the meaning of a number's “representation” on Wolfram Alpha?

When searching a number on Wolfram Alpha, one of the results is its representation.
For example, for 8549:

8549 has the representation 8549 = $5·2^6·3^3-91$.

Similarly for 75290:

75290 has the representation 75290 = $3·2^9·7^2+26$.

What is the significance of these representations?

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Hum, it shows the numbers as sums or differences of products of small primes. It is curious. – Willie Wong Jan 18 '11 at 14:39
Have you tried to ask Wolfram Alpha? If not, you could send an e-mail to this adress: info@wolframalpha.com or start a discussion at their forum: community.wolframalpha.com – Max Muller Jan 18 '11 at 19:37
My guess is that it is the "shortest" way of expressing this number. Note that it doesn't do it for primes. – Noon Silk Jan 18 '11 at 23:18
Very well. I've submitted a question to their forum, I hope to get results. I wanted to give Math.SE the first chance of ansering the question. It is possible that line is just a filler... – Kobi Jan 19 '11 at 5:12
@Noon Silk - Well, isn't the "shortest" way simply writing the number? 8549 is quite short. And 83×103 (prime factors) is also shorter than that representation. – Kobi Jan 19 '11 at 5:17

2 Answers

What it seems to do is, when $n$ is your number, that it maximizes the number of prime factors of $q$ within the range $q \in (n-100,n+100)$. And then sets $n=q+(n-q)$. Doing this it can easily find that 513 is for example $513=2^9+1$. However for the numbers you gave it is not really interesting.

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@user3123: is this a guess, or is this referred to in some documentation? (In particular, why $(n-100,n+100)$?) And what is the criterion for making the result "noteworthy" enough to be displayed? Note that for many large numbers the properties section lists no such representations. – Willie Wong Jan 18 '11 at 14:43
@Willie It is a guess and the $\pm100$ comes because I tried like 50 numbers and what it added was always below $100$ but sometimes near to $100$. Also I see that my initial guess is at least not fully correct. Consider that $9660=3*2^7*5^2+60$ but Wolfram writes it as $9660=7^4*2^2+56$. Using only $6$ factors instead of the optimal $10$. – Listing Jan 18 '11 at 17:18

I think it is just a curious fact to know and tell. They seem to be a product of small primes plus or minus a small correction. For 2010, besides the "obvious" 2010=2^11-38 it also finds that 2010 divides 29^6-1.

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I would argue it isn't a curious fact at all. I'd say that without any noticeable pattern, it's no more or less interesting than any other random expression that equates to the same value. – Michael J Swart Oct 9 '13 at 19:44