Reddit mentions of Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication

For those curious, an integer triple (a,b,c) makes a right triangle with hypotenuse c, and a & b with no common factors, exactly when there are two other integers m and n with no common factors, and one of them even, so that

• a=m^(2)-n^(2)
  
• b=2mn
  
• c= m^(2)+n^(2)
  
  This means if you have any two numbers, m and n, that satisfy these conditions, then you can generate an integral triangle via this formula.This is the main generating formula for Pythagorean Triples.
  
  But then the question becomes "Which numbers can be a hypotenuse in an integer triangle?" Because of this formula, this is the same as answering "Which numbers are the sums of two squares?" The above formulas were discovered by Euclid in 200BC, but the answers to these other questions were answered ~2000 years later by Euler.
  
  As it turns out, we can actually only worry about which odd prime numbers can be a hypotenuse. If we can answer "Which prime numbers are the sum of two squares?" then we can answer it for any number through some basic manipulations. The answer is, maybe surprisingly, "A prime is a sum of two squares exactly when it has remainder 1 when divided by 4." This means that a prime number being a hypotenuse is equivalent to how it looks when we divide it by 4! Crazy!
  
  For example, 5 can be a hypotenuse with legs of length 3 and 4. Just take m=1 and n=2 and apply the formulas. But 7 cannot be a hypotenuse because it has remainder 3 when divided by 4.
  
  This is the first case of Quadratic Reciprocity and is a beginning step in establishing what is now known as Class Field Theory which tells us a whole lot about certain types of polynomials. The full power of Class Field Theory is required to find out which prime numbers have the form x^(2)+ny^(2), Euler's case was when n=1. It took about 200 years from the time of Euler to the completion of Class Field Theory, by Teiji Takagi and Emil Artin. The idea of Class Field Theory is that Apples (Hypotenuses of triangles) and Oranges (dividing by 4) are actually the same thing.
  
  But Class Field Theory asked more questions than it answered. There are many more classes of polynomials that cannot be characterized by Class Field Theory. The main attempt at solving this problem is Langlands Program which is currently a huge effort in today's math. It is very very hard but relates to most every field of math and even has implications in physics. Langlands takes the idea of Apples and Oranges being the same thing to a whole new level. It says that there is really no difference between the entire evolutionary history of Apples and Oranges.
  
  EDIT: Wow, gold! Thanks :)

I've seen something similar to this used to evaluate modular functions. I don't know the details but the idea is very cool:

Let

[; \displaystyle{\gamma_{2}(\sqrt{-m})=\mathfrak{f}_{2}(\sqrt{-m})^{16}+\frac{16}{\mathfrak{f}_{2}(\sqrt{-m})^8}} ;]

and

[; \displaystyle{\mathfrak{f}_{2}(\sqrt{-m})=\sqrt{2}q^{1/24}\prod_{n=1}^{\infty}(1+q^n)} ;]

where m = 1,2,4,7 and [; q=e^{-2\pi\sqrt{m}} ;]. Then it can be shown, presumably using some number-theoretic wizardry, that [; \gamma_{2}(\sqrt{-m}) ;] is an integer. Estimating the infinite product is easy:

[; \displaystyle{1<\prod_{n=1}^{\infty}(1+q^n)<\prod_{n=1}^{\infty}e^{q^n}=e^{q/(1-q)}} ;]

and since [; q\leq e^{-2\pi} ;], [; \displaystyle{\frac{q}{1-q}\leq\frac{q}{1-e^{-2\pi}}<1.002q} ;]. Substituting this in:

[; 256q^{2/3}+q^{-1/3}e^{-8.016q}<\gamma_{2}(\sqrt{-m})<256q^{1/3}e^{16.032q}+q^{-1/3} ;]

Then using [; \displaystyle{1-e^{-x}<\frac{x}{1-x}} ;] the upper bound minus the lower bound is at most

[; \displaystyle{256q^{2/3}(e^{16.032q}-1)+\frac{8.016q^{2/3}}{1-8.016q}} ;]

which is an increasing function. Since [; q\leq e^{-2\pi} ;], this is at most 0.2413.... Hence, [; \gamma_{2}(\sqrt{-m}) ;] can be computed exactly by evaluating this approximation and rounding it to the nearest integer.

source

You should be aware that there is a second edition, which is much more reasonably priced. It really is a great book, you're in for a treat!

I unfortunately don't know of a good generally accessible source on this. It's really one specific (and particularly nice looking) example of a more general theory. As such it's the sort of thing you usually learn as an example while you're learning about the more arithmetic side of modular forms, and I don't know of any good self contained sources for it.

What's your current background in math? If you haven't already taken algebraic number theory, you'd definitely want to start there (although you'd need abstract algebra, and especially Galois theory, as a prerequisite). The most important concept to pick up there would be the notion of a "Frobenius element".

If you're already familiar with algebraic number theory, then (as other people have said here), primes of the form x^(2)+ny^(2) by Cox is a good place to start, although if I remember correctly they don't actually do the n = 23 example there. That combined with a book on modular forms, such as Diamond and Shurman should give you a pretty good understanding of this.