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I am a total newbie to the world of math and was interesting in learning. I just finished my degree(non-math) and am going to study a few math books to see if it interests me to apply for something more quantitative but I want to study something interesting with interesting problems that won't bore me.

I thought about it, and thought of the type of problems that intrest me. One is predicting the future and the other is predicting the past. Here's a problem that I think would be cool. Say you have a list

calories 89, 34, 67, 43, 54, 232, 623

and someone tells you that someone had a total of "6553" calories in a day. What type of math would try to figure this out? Is it algebra? (by the way to get this question all I did was take each value above and multiple first one by 1, second one by two, etc.up to 7.)

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    I'm not sure what you're asking. What are you trying to figure out exactly? How many calories someone is likely to have in a day given a list of how many calories they've had per day recently? That sounds to me like statistics.2011-03-18
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    Never mind - Arturo cleared it up for me, and I concur with him that this is number theory.2011-03-18

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To cast the problem a little more clearly, you have a number of "weights", $w_1,\ldots,w_n$, in this case: \begin{align*} w_1 &= 89\\ w_2 &= 34\\ w_3 &= 67\\ w_4 &= 43\\ w_5 &= 54\\ w_6 &= 232\\ w_7 &= 623, \end{align*} and a "target total" $T$, in this case $T=6553$. You want to find nonnegative integers $a_1,\ldots,a_n$ such that $$a_1w_1 + \cdots + a_nw_n = T.$$

In its broadest sense, this is an example of what is called a Diophantine equation (an equation in which we require the solutions to be nonnegative integers, or more generally rational numbers). They are studied in the branch of mathematics called Number theory.

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    thank you so much, Arturo. I'm wondering in number theory would a question like this be solvable? is it practical or abstract theory?2011-03-18
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    @Lostsould: Although Diophantine equations, *in general* can be pretty hard (Fermat's Last Theorem is an example of a Diophantine equation, for instance), this particular kind of problem (a "linear diophantine equation") is generally straightforward (in a practical sense). There are well-known, good algorithms for solving them (or showing no solution exists). But even a small tweak can make the problem very hard; see for example the closely related knapsack problem (http://en.wikipedia.org/wiki/Knapsack_problem) and postage stamp problem (http://en.wikipedia.org/wiki/Postage_stamp_problem)2011-03-18
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    Thank you so much for your quick answers Arturo. One last question. Do you think its feasible for someone to learn number theory directly? or should they learn something else..My math isn't amazing but I do have a computer background and worked in investment management(not crazy number crunching but I think I'm a quick learner). If there's a better way to get into number theory then what would you suggest that is?2011-03-18
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    @Lostsoul: Elementary number theory is both beautiful, classical, and full of interesting stuff without needing too much background. It's a great gateway to "bigger and better" things (leads to both complex analysis and to abstract algebra). Advanced techniques would require more background, but one can go pretty far regardless.2011-03-18
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    @Lostsoul: Also, I think I misinterpreted your second question in the first comment. Number theory has turned out to have amazing practical applications that nobody even suspected as recently as 100 years ago. For example, it is a key component of internet security.2011-03-18
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    Wow thank you so much Arturo. You sold me! It seems very valuable to reverse engineer. Couldn't it be used to figure out what competitors are doing if you have their production numbers. Or if a company reports they earned XX amount of money a year, this seems like it could break down how many of each of their products they sold? This is truly amazing. I wonder why I finished a computer degree and just finished a business degree and never heard of this before.2011-03-18
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    @Lostsoul: Note that even a simple problem like the one you give *could* have more than one solution. So there may only be limited amounts of information you could derive from knowing the totals.2011-03-18
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    @Arturo Thanks..are the limitions of number theory really great? It seems so interesting and so powerful yet I wonder why it doesn't appear to wide spread. I used to work at an investment management firm and I can see many ways this could be applied to assisting research analyst and companies make better predictions by understanding the past better. I really hope I can apply it to real world problems, but if not it still seems so interesting to learn from.2011-03-18
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    @Lostsoul: There are problems that are very hard to solve *practically*. For example, factoring a big number is easy in principle (just try dividing all the primes up to $\sqrt{n}$ to see if it has a factor, then lather, rinse, and repeat), but for large numbers it can be so computationally expensive that you wouldn't be able to carry out to procedure before the sun burns out. Other problems admit too many possible solutions that they don't let you get much information. Not so much a limitation of number theory, but rather of what one wants to actually be able to do.2011-03-19
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    I'm sorry i don't want to keep on bugging you on this site, but your great answers are making me think and creating other questions. Do you think the emergence of cloud computing and more powerful cpu power solving the the big problems to make them more practical? I mean if you look at google's systems they process billions and billions amounts of data every time we search within a mili seconds..does number theory get better with a larger richer data set or does it complicate the results?2011-03-19
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    @Lostsoul: Parallelizing (using several computers working together) only goes so far. It does push the need for larger problems (RSA keeps producing bigger and bigger primes for their security systems). For most of these problems, whether one can solve them in a practical way is a question of the size of the problem; so far as we know, the difficulty grows exponentially (or sub-exponentially) with the size of the problem. Our computing capacity has no hope of keeping up. Even accepting Moore's law, we can double the size of the problem in a lot less than 2 years.2011-03-19
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    Sorry I should have marked this resolved a while ago. I have been a bit busy and am just starting to venture into this project, I have done a little reading and I can't thank you enough for introducing me to this, Arturo. I have a book on the knapsack problem and will eventually look at the diophantine equation. Thank you so much Arturo(I will def be more active and bugging you all with questions as I dig into the material).2011-05-09
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For your question you need to find positive integer solutions to $89a +34b +67c +43d+ 54d +232d+ 623e=6553$, $\{a,...,e\}$ are the number of items you eat of each to get 6553 calories. Problems like this where you need to find integer solutions are called linear Diophantine equations. But you can treat them as puzzles and try to use some hit and trial methods/computations. Sometimes, though not always, Wolfram alpha might help.