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While reading popular science book Fermat's Last Theorem I was amazed to find out that in number theory interesting things happen even at very large scales. For instance the Graham's number was named "the largest number ever used in a serious mathematical proof", but this is quite rare (is it?) and it seems that lot of the "interesting" numbers are located not far from zero (Should I say in the middle of the number line? Or at the center of the complex plane?)

I am not a mathematician, I don't even know how to ask the right thing and my question can be closed as off topic. But I hope someone will understand what I mean and reply with interesting thoughts or recommended reading.

So, what is the "shape" of numbers in Number Theory?

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I don't know what you mean by shape, but this previous question seems in the spirit of what you're asking: Why are all the interesting constants so small? –  Rahul Jan 7 '13 at 18:30
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Not directly related to your question, but Benford's Law (en.wikipedia.org/wiki/Benford%27s_law) is one fascinating result about the "shape" of numbers that occur in data. –  user7530 Jan 7 '13 at 18:40
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It's interesting to note that numbers that are extremely close to $0$ are also rarely used. How many mathematical works include specific mentions of numbers on the order of $10^{-100}$? –  Shaun Ault Jan 7 '13 at 18:49
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In fact, it seems that the the set of numbers $\{ 1/x \;|\; x \; \textrm{is "interesting"}\}$ should be almost indistinguishable from the set of "interesting" numbers, whatever the term "interesting" means. –  Shaun Ault Jan 7 '13 at 18:51
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Maybe this: The plan is for this to be an introductory textbook on elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. The book might have been called "Geometry of Numbers" except that this already has an established meaning different from what we have in mind here. From Hatcher's website math.cornell.edu/~hatcher/TN/TNpage.html –  Sigur Jan 7 '13 at 20:46

2 Answers 2

There are the figurate numbers (triangular numbers, square numbers, etc.)

I have my own idiosyncratic definition of a shape of a natural number (I use it with my number theory students). I use the term to refer to the form of the prime factorization. So for instance the shape of $12$ is $p^2q$. The shape of $18$ is also $p^2q$ (I list shape starting with the highest power of any prime). This notion of shape tells you interesting things about the number such as how many factors it has. We can ask questions like what shapes of numbers have exactly $20$ factors. Also shape gives a nice equivalence relation on the natural numbers.

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I also don't understand what's meant by "shape" here.

But it wouldn't be hard for small groups of integers to be assigned to shapes - either the familiar way Descartes did or (differently) as some Indian schoolteachers traditionally do to teach the multiplication tables using 2D forms or shapes.

Some of these shapes in Indian school maths are superimposed on grids of numbers and provide good ways to visualise distributions or moduli of certain multiples as layouts.

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