Solving the Equation (a^2 - b^4 2009)

In this article, we will delve into solving the equation (a^2 - b^4 2009) where a and b are positive integers. This involves understanding the constraints and utilizing mathematical tools to find a feasible solution.

Understanding the Problem

The given equation is:

(a^2 - b^4 2009)

To solve this, we can start by rearranging the equation to express (a^2) in terms of (b^4):

(a^2 2009 b^4)

Since a and b are positive integers, (b^4) must be less than 2009. This means we need to find the maximum integer value of b which is the fourth root of 2009:

(b approx sqrt[4]{2009} approx 6.7)

Hence, b can take values from 1 to 6. We will evaluate each possible value of b to check if (a^2) is a perfect square.

Evaluating Possible Values of b

Let's evaluate the equation for each integer value of b from 1 to 6:

For b 1: (a^2 2009 - 1^4 2008) 2008 is not a perfect square. For b 2: (a^2 2009 - 2^4 2009 - 16 1993) 1993 is not a perfect square. For b 3: (a^2 2009 - 3^4 2009 - 81 1928) 1928 is not a perfect square. For b 4: (a^2 2009 - 4^4 2009 - 256 1753) 1753 is not a perfect square. For b 5: (a^2 2009 - 5^4 2009 - 625 1384) 1384 is not a perfect square. For b 6: (a^2 2009 - 6^4 2009 - 1296 713) 713 is not a perfect square.

Since none of the calculated values of (a^2) are perfect squares for b from 1 to 6, we conclude that there are no positive integer solutions to the equation (a^2 - b^4 2009).

Alternative Solution Using Factorization

It turns out that the equation can be factorized as:

[(a - b^2)(a b^2) 41 times 49]

By examining the factors of 2009, we find that a possible solution is:

[begin{align*}a - b^2 41a b^2 49end{align*}]

By solving these two equations simultaneously:

[begin{align*}a - b^2 41 quad text{(Equation 1)}a b^2 49 quad text{(Equation 2)}end{align*}]

Adding Equation 1 and Equation 2:

[(a - b^2) (a b^2) 41 49]

(2a 90)

(a 45)

Substituting (a 45) into Equation 1:

[(45) - b^2 41]

(-b^2 -4)

(b^2 4)

(b 2)

Hence, the solution is a 45 and b 2. Therefore, the value of (ab) is:

(ab 45 times 2 90)

Conclusion

The alternative solution method using factorization reveals that the positive integer solutions to the equation are a 45 and b 2, giving (ab 90).