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Some number theory basics: A minithread
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This thread is going to be very basic, and many people here know this. But maybe not all! And they may still be interested, so that's why I'm making this thread. Despite that, I'm just gonna say I might still make mistakes. With that out of the way, here goes!
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So in number theory, we study whole numbers. Usually that should mean natural numbers but that's hard, so let's say instead integers. But, even though we are interested only in integers, we may need to study things that aren't integers, to help us understand the integers.
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What do I mean? Recall the formula for difference of squares:
a^2-b^2=(a+b)(a-b)
It's a wonderful formula, but what about sums of squares? People have been interested about this since ancient times.
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a^2-b^2=(a+b)(a-b)
It's a wonderful formula, but what about sums of squares? People have been interested about this since ancient times.
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To factor a sum of squares, we need complex numbers:
a^2+b^2=(a+ib)(a-ib)
This remains one of the most beautiful things in math to me.
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a^2+b^2=(a+ib)(a-ib)
This remains one of the most beautiful things in math to me.
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Though I haven't elaborated much yet, I hope this gives a taste of the kind of strategies we use in number theory - by making things more complicated (in this case we introduced complex numbers), we make studying simple but difficult things easier.
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Now I want to introduce some basic terms in number theory. The complex numbers of the form a+bi where a and b are *integers* are called the "Gaussian integers". We will use them in this thread to illustrate basic concepts in number theory.
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The basic building blocks of the integers are the primes. Note that we can talk about primes only for integers - in the rationals everything (except zero) divides everything else, so "primes" make no sense. In the Gaussian integers, there are also primes!
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But these primes are different from the primes of the ordinary integers! For example, consider 5 in the Gaussian integers. 5 can now be factored into (1+2i)(1-2i), and hence is *not* prime! 3 is still prime, however. We say that 5 "splits", while 3 "remains prime".
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By what we described earlier, 5 is a sum of two squares: 5=1^2+2^2. Again, the point is that all this complicated machinery relates back to simple stuff. 3, on the other hand, is not a sum of two squares.
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It will take more machinery to determine which Gaussian integers are prime and which one are not. Furthermore, there are certain complex numbers for which the concept of "prime" needs to be redefined! But this requires notions of abstract algebra.
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Instead, I want to focus on the peskiest of the primes, 2, the smallest and the only one that is even. Does it split in the Gaussian integers? Does it remain prime? Turns out it's neither!
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In the Gaussian integers, we have 2=i(1-i)^2. We don't count the i, just like we don't count -1 when factoring - otherwise we could have infinitely many factorizations that mean the same thing, because (-1)^2=1, and any number times 1 is itself.
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The number i, like -1, is a "unit", and we ignore them when factorizing. So ignoring i, 2 is a square in the Gaussian integers. We don't say it "splits" or "remains prime". Here's a word you might see a lot in math - we say it "ramifies".
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So "splits", "remains prime", "ramifies". These are the three things that could happen to an ordinary prime number when we look at them as Gaussian integers. And they tell us a lot about integers. That's the end of this minithread, hopefully more to come soon!
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