Given the equation $a^2 + b^2 = 2809$, there are multiple pairs of values for $a$ and $b$ that can satisfy this equation. To find integer values for $a$ and $b$, we can start by factoring 2809, or checking values for $a$ and solving for $b$.

1. Take a value of $a$, for example $a = 35$.

   $a^2 = 35^2 = 1225$

2. Substitute this into the original equation:

   $a^2 + b^2 = 2809 \implies 1225 + b^2 = 2809$

   $b^2 = 2809 - 1225 = 1584$

Since $b^2 = 1584$, this is not a perfect square, so $a = 35$ is not a solution. We continue testing different integer values for $a$.

Trying another value, say $a = 37$:

$a^2 = 37^2 = 1369$

Substitute into the equation:

$1369 + b^2 = 2809$

$b^2 = 2809 - 1369 = 1440$

Again, 1440 is not a perfect square.

Now, let's try $a = 53$:

$a^2 = 53^2 = 2809$

In this case:

$2809 + b^2 = 2809 \implies b^2 = 0$

Thus, $b = 0$, and the solution is $a = 53$, $b = 0$.

Therefore, one possible solution is $a = 53$ and $b = 0$.

Would you like to explore other solutions or have any questions?

Here are 5 related questions to expand on this concept:

1. What are the conditions for a number to be a perfect square?
2. Can there be negative values for $a$ or $b$ in this problem?
3. How can we determine all pairs of integers that satisfy $a^2 + b^2 = 2809$?
4. What is the significance of the Pythagorean theorem in equations like this?
5. How can we solve quadratic Diophantine equations in general?

Tip: When solving equations like $a^2 + b^2 = N$, it's helpful to check for perfect squares and use factoring techniques!