Problem: Find the number of lattice points on the curve 1/x - 1/y = 1/2013.

Solution:

To find the number of lattice points on the given curve, we need to find all the pairs of positive integers (x, y) that satisfy the equation 1/x - 1/y = 1/2013.

Step 1: Simplify the equation
Multiply both sides of the equation by 2013xy to eliminate the denominators:

2013y - 2013x = xy

Step 2: Rearrange the equation
Rearrange the equation to bring all the terms to one side:

xy + 2013x - 2013y = 0

Step 3: Apply Simon's Favorite Factoring Trick
To solve this equation, we can use Simon's Favorite Factoring Trick. We add and subtract a constant term to create a perfect square trinomial:

xy + 2013x - 2013y + 2013^2 = 2013^2

Now, we can rewrite the equation as:

(x + 2013)(y - 2013) = 2013^2

Step 4: Factorize 2013^2
The prime factorization of 2013^2 is 3^2 * 11^2 * 61^2. Therefore, there are 3 * 3 * 3 = 27 factors of 2013^2.

Step 5: Count the number of lattice points
For each factor of 2013^2, we can find a corresponding pair of positive integers (x, y) that satisfies the equation. Since (x + 2013)(y - 2013) = 2013^2, we can set x + 2013 = factor and y - 2013 = 2013^2 / factor.

For example, if factor = 3, then x + 2013 = 3 and y - 2013 = 2013^2 / 3. Solving these equations, we get x = -2010 and y = 2020421.

We repeat this process for all 27 factors of 2013^2 and find the corresponding pairs of positive integers (x, y).

Step 6: Count the number of lattice points
For each pair of positive integers (x, y), we check if x and y are both positive. If they are, we count it as a lattice point on the curve.

Finally, we add up all the lattice points obtained from Step 5 to get the total number of lattice points on the curve 1/x - 1/y = 1/2013.

Conclusion: The value of "k" (the number of lattice points on the curve) can be determined by following the steps mentioned above. Since the prime factorization of 2013^2 has 27 factors, there are 27 lattice points on the curve 1/x - 1/y = 1/2013.