# 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 http://www.research.att.com/ njas/sequences/A002163sequence 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.txthttp://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.pdfhttp://eprints.maths.ox.ac.uk/32/01/wilson.pdf
Title square root of 5 SquareRootOf5 2013-03-22 17:28:12 2013-03-22 17:28:12 MathNerd (17818) MathNerd (17818) 6 MathNerd (17818) Definition msc 11A25