## You are here

Homesquare root of 5

## Primary tabs

# square root of 5

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.

# References

- 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

## Mathematics Subject Classification

11A25*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb