I'm looking for interesting and/or expressive quotations from mathematicians about the class number formula. I'm interested both in quotations from historical mathematicians and from modern mathematicians. I'm also interested in quotations about generalizations of the class number formula.

I'll start off by giving one, paraphrasing from Henri Darmon and Claude Levesque's article titled Infinite sums, diophantine equations and Fermat's Last Theorem (pages 4-6):

Let $N_p$ be the number of solutions to $x^2 + y^2 = 1$ over $\mathbb{F_p}$, let $N_{\mathbb{Z}}$ be the number of solutions over $\mathbb{Z}$ and let $N_\mathbb{R}$ be the circumference of the circle. From quadratic reciprocity, Leibniz's formula, and the Euler product formula we deduce $\displaystyle \prod_{p} > \frac{N_p}{p} = \frac{4}{\pi}$. We conclude that $\displaystyle \prod_{p} > \frac{N_p}{p} \cdot N_{\mathbb{R}} = > 2N_{\mathbb{Z}}$. This magical formula shows that the numbers $N_p$ "know" the behavior of the equation over the real numbers. Fundamentally, this is only a simple reinterpretation of Leibniz's formula, but in fact this is quite a fruitful one.

  • $\begingroup$ Sorry, I don't see where any class number formula comes into the quote you give. $\endgroup$ – Gerry Myerson Nov 15 '12 at 22:02
  • 1
    $\begingroup$ It's the special case $\mathbb Q(i)$, class number 1. $\endgroup$ – Stopple Nov 15 '12 at 23:27

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