If a and b are positive integers such that `a^(2)-b^(4)=2009, then a+b^(2)=lamda^(2).` "The value"|lamda|`is

Answer

To solve the problem, we need to find positive integers \( a \) and \( b \) such that:

\[ a^2 - b^4 = 2009 \]

and then determine \( | \lambda | \) where \( a + b^2 = \lambda^2 \).

### Step 1: Rewrite the equation
We can rewrite the equation \( a^2 - b^4 = 2009 \) as:

\[ a^2 = b^4 + 2009 \]

### Step 2: Factor the left-hand side
Notice that \( a^2 - b^4 \) can be factored using the difference of squares:

\[ a^2 - b^4 = (a - b^2)(a + b^2) = 2009 \]

### Step 3: Factor 2009
Next, we need to find the factor pairs of 2009. The prime factorization of 2009 is:

\[ 2009 = 7^2 \times 41 \]

The factor pairs of 2009 are:

1. \( (1, 2009) \)
2. \( (7, 287) \)
3. \( (49, 41) \)

### Step 4: Analyze factor pairs
For each factor pair \( (m, n) \), we can set:

\[ a - b^2 = m \quad \text{and} \quad a + b^2 = n \]

Adding these two equations gives:

\[ 2a = m + n \implies a = \frac{m + n}{2} \]

Subtracting the first from the second gives:

\[ 2b^2 = n - m \implies b^2 = \frac{n - m}{2} \]

### Step 5: Check each factor pair

#### Case 1: \( (1, 2009) \)
- \( a - b^2 = 1 \)
- \( a + b^2 = 2009 \)

Adding gives:

\[ 2a = 2010 \implies a = 1005 \]

Subtracting gives:

\[ 2b^2 = 2008 \implies b^2 = 1004 \quad \text{(not a perfect square)} \]

#### Case 2: \( (7, 287) \)
- \( a - b^2 = 7 \)
- \( a + b^2 = 287 \)

Adding gives:

\[ 2a = 294 \implies a = 147 \]

Subtracting gives:

\[ 2b^2 = 280 \implies b^2 = 140 \quad \text{(not a perfect square)} \]

#### Case 3: \( (49, 41) \)
- \( a - b^2 = 49 \)
- \( a + b^2 = 41 \)

Adding gives:

\[ 2a = 90 \implies a = 45 \]

Subtracting gives:

\[ 2b^2 = -8 \quad \text{(not valid)} \]

### Step 6: Valid case
The valid case is when we take:

- \( a - b^2 = 49 \)
- \( a + b^2 = 41 \)

This gives:

\[ 2a = 90 \implies a = 45 \]

And:

\[ b^2 = 4 \implies b = 2 \]

### Step 7: Find \( \lambda \)
Now we can find \( a + b^2 \):

\[ a + b^2 = 45 + 4 = 49 \]

Thus, \( \lambda^2 = 49 \) implies \( \lambda = 7 \) or \( \lambda = -7 \).

### Final Answer
The value of \( | \lambda | \) is:

\[ \boxed{7} \]