2000 AIME II Problem 2

Question

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola

x^2 - y^2 = 2000^2?

Answer

Factor

x^2 - y^2 = (x - y)(x + y)

and note the two factors have the same parity

Write

x - y = 2a

and

x + y = 2b

with

ab = 10^6;

count ordered pairs, and remember negative values of

x

Factor

(x - y)(x + y) = 2000^2

= 2^8 \cdot 5^6.

The factors

x - y

and

x + y

have the same parity, and their product is even, so both must be even. Writing

x - y = 2a

and

x + y = 2b

gives

ab = 2^6 \cdot 5^6 = 10^6.

Each ordered pair of positive integers

(a, b)

with

ab = 10^6

yields exactly one solution

x = a + b,

y = b - a

with

x \gt 0,

and

10^6

has

7 \cdot 7 = 49

divisors, hence

49

such pairs. Replacing

(a, b)

by

(-a, -b)

gives the

49

solutions with

x \lt 0,

and

x = 0

is impossible since

-y^2 \lt 2000^2.

In total there are

49 + 49 = 98

lattice points.

Problem 2 in other years