000RM.dvi

2.2 Lattice points on a circle 111

2.2 Lattice points on a circle ...........

How many lattice points are there on the circlex^2 +y^2 = n^2? This
number is usually denoted byr 2 (n^2 ). The first few values can be read
from the following figure.

n 1 2 3 4 5 6 7 8 9 10

r 2 (n

2

)

In general, if the numbernis factored into a product of prime powers

n=2a·

∏

i

pbii·

∏

j

q

cj

j,

wherepiandqj are prime numbers of the forms 4 k+1and 4 k+
respectively, then

r 2 (n^2 )=

∏

i

(2bi+1).

Thus, for example, the circle of radius 100 has lattice points. On
the other hand, ifpis a prime number of the form 4 n+1, then the circle
of radiusphas 12 lattice points. Four of these are on the coordinate axes.
The other 8 are of the form(±a,±b),(±b,±a), and depend on how the
primepis written as a sum of two squares.