Favorite Number Theory Fact

Here's my favorite number theory fact I've come across so far - but that's probably just because it's the one I've proved most recently.

1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2

In English, the sum of the first n consecutive cubes is always equal to the square of the sum of the bases of those cubes.  How crazy is that :)

Atrytone Atrytone
22-25, F
6 Responses Feb 24, 2010

@Atrytone: Euclid. But there <i>is</i> a last prime number that I will personally experience or will be known during my lifetime, and there may indeed be a last prime number that the Universe will experience (whatever that means), even if its existence is infinite. So it depends on how you define "last".<br />
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My intuition is that the Universe (and hence my own existence) depends upon either the prime numbers or something which leads to the prime numbers (Reimann? Zeta?). The nature of that connection is something we may discover. I have my own conjectures about that, starting from the interesting observation ("fact") that the multiplicative inverse (division) is only well-defined in modular arithmetic over the set of prime numbers.<br />
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The concept of "last" itself only makes sense if you start from a finite point of view and reason toward an infinite one. The vast majority of mathematics has that bias.<br />
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It seems to me that we should be starting with infinite concepts and seeing what footprint those algorithms or ex<x>pressions generate when applied (e.g., iterated n=1 to infinity over the Integers). Some ex<x>pressions (such as "n") exactly result in the Integers, some in the Reals, some in the Primes, some in other sets named or unnamed, and so on. <br />
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The word "ex<x>pression" used to refer to an algebraic formulation has always seemed to be inverted to me. Isn't the result of iteration over some formalized concept its "ex<x>pression"? But what then should we call the formalized concept?<br />
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(By the way, what set would you propose be named the "miracle" numbers? Or perhaps the "miracle" primes? What generates or "recognizes" that set? Or do you think that every prime is a little miracle?)

@NewChrissy: Have you seen the proof that there's no last prime number?

@freeed: How could there NOT be any prime numbers? I don't find their existence all that interesting, after all the definition is pretty simple. It is all of their other properties & relationships that are surprising. And how they keep showing up unexpectedly. Clearly those relationships are not "accidental", they suggest that something fundamental is going on. But what?

Why are there prime numbers? There wouldn't be, if I'd created them. It works because it does.<br />
How, if at all, does the Axiom of Choice affect number theory; that's another intriguing tidbit.

Just for fun :) Or, rather, partly because it's an intriguing pattern and partly because I wanted to conquer the challenge.<br />
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Even having proved it, it's still pretty mysterious to me. WHY DOES IT WORK? :D

That is TOTALLY COOL! How beautiful this branch of math is. "numbers and mathematics are the vibrations of God".<br />
Did you prove it as a math problem in a course or "just for fun"?.