MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

3 added 1 characters in body

I learned this example from Noam Elkies's excellent article The Klein Quartic in Number Theory. Mazur Elkies observes that Siegel's 1968 paper Zum Beweise des Starkschen Satzes, in order to prove its main result, proves what is equivalent to the following.

Theorem: Suppose that the only Fibonacci numbers which are cubes are $0, \pm 1, \pm 8$. Then the set of negative integers $d$ such that $\mathbb{Q}[\sqrt{d}]$ has class number $1$ is $\{ -1, -2, -3, -7, -11, -19, -43, -67, -163 \}$.

2 edited body

I learned this example from Barry Mazur's Noam Elkies's excellent article The Klein Quartic in Number Theory. Mazur observes that Siegel's 1968 paper Zum Beweise des Starkschen Satzes, in order to prove its main result, proves what is equivalent to the following.

Theorem: Suppose that the only Fibonacci numbers which are cubes are $0, \pm 1, \pm 8$. Then the set of negative integers $d$ such that $\mathbb{Q}[\sqrt{d}]$ has class number $1$ is $\{ -1, -2, -3, -7, -11, -19, -43, -67, -163 \}$.

1

I learned this example from Barry Mazur's excellent article The Klein Quartic in Number Theory. Mazur observes that Siegel's 1968 paper Zum Beweise des Starkschen Satzes, in order to prove its main result, proves what is equivalent to the following.

Theorem: Suppose that the only Fibonacci numbers which are cubes are $0, \pm 1, \pm 8$. Then the set of negative integers $d$ such that $\mathbb{Q}[\sqrt{d}]$ has class number $1$ is $\{ -1, -2, -3, -7, -11, -19, -43, -67, -163 \}$.