To solve the equation \(2^x - 2^y = 496\), we need to find integer values for \(x\) and \(y\) such that the equation holds true.

First, let's rewrite the equation in a more convenient form:
\[ 2^x - 2^y = 496 \]

We can factor out \(2^y\) from the left-hand side:
\[ 2^y (2^{x-y} - 1) = 496 \]

Now, we need to find integer values for \(x\) and \(y\) such that \(2^y\) and \(2^{x-y} - 1\) multiply to 496. Let's start by examining the factors of 496.

The prime factorization of 496 is:
\[ 496 = 2^4 \times 31 \]

This suggests that \(2^y\) could be \(2^4 = 16\) and \(2^{x-y} - 1\) could be 31. Let's test this hypothesis:

If \(2^y = 16\), then:
\[ y = 4 \]

Now, we need \(2^{x-4} - 1 = 31\):
\[ 2^{x-4} = 32 \]
\[ x - 4 = 5 \]
\[ x = 9 \]

So, the solution is:
\[ x = 9 \]
\[ y = 4 \]

Let's verify the solution:
\[ 2^9 - 2^4 = 512 - 16 = 496 \]

The values \(x = 9\) and \(y = 4\) satisfy the equation. Therefore, the solution to the equation \(2^x - 2^y = 496\) is:
\[ x = 9 \]
\[ y = 4 \]