The square root of 5 is an irrational number involved in the formula for the golden ratio. It is also used in statistics when dealing with 5-business day weeks. Its decimal expansion begins 2.2360679774997896964, see sequence A002163 in Sloane's OEIS. Its simple continued fraction is 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... One formula for the square root of 5 involves some of the same numbers as in Euler's identity (but with a 2 instead of the 1): $e^{i\pi}+2\phi$ . The square root of 5 modulo a prime number is employed in some ECM algorithms.

A rectangle with unit height and $\sqrt{5}$ width can be split into two golden rectangles of the same size and a square, or into two golden rectangles of different sizes.

The conjecture stating ``that any abelian surface with RM by $Q(\sqrt{5})$ is isogenous to a simple factor of the Jacobian of a modular curve $X_0(N)$ for some $N$ '' still stands. John Wilson has produced equations for curves of genus 2 with Jacobians of the specified RM.

Bibliography

1

Francois Morain. Primality Proving Using Elliptic Curves: An Update. Springer: Berlin (2004)

2

Robert Nemiroff and Jerry Bonnell. A million digits of sqrt(5) at Project Gutenberg http://www.gutenberg.org/dirs/etext96/5sqrt10.txt

3

Clifford Pickover. Wonders of Numbers, Oxford: Oxford University Press (2001) p. 106.

4

John Wilson ``Curves of genus 2 with real multiplication by a square root of 5'' p. i Dissertation, Oxford University, Oxford (1998) http://eprints.maths.ox.ac.uk/32/01/wilson.pdf