Spiral of Primes

a

1

4

9

b

1

c

41

R

16

The natural numbers are arranged along an Archimedean spiral so that the squares 1, 4, 9, … are on the

x

axis, with the primes drawn in blue, up to a circle of radius

R

. The spiral is parametrized in polar coordinates by

r=

n

,

θ=

n

. Also shown is the curve given by

r=

a

2

n

+bn+c

,

θ=r-n

a

(red), and its intersections with the spiral (purple boxes).