Find the number of quadruplets of positive integers (a,b,c,d) satisfying the following relations . `1 le a le b le c le d and ab + cd = a + b + c + d + 3`

Answer

To solve the problem of finding the number of quadruplets of positive integers \((a, b, c, d)\) that satisfy the conditions \(1 \leq a \leq b \leq c \leq d\) and \(ab + cd = a + b + c + d + 3\), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation given:
\[ ab + cd = a + b + c + d + 3 \]
Rearranging this gives us:
\[ ab + cd - a - b - c - d = 3 \]

### Step 2: Introducing Variables
To simplify the analysis, we can introduce new variables. Let's denote:
- \(a = x\)
- \(b = x + k_1\)
- \(c = x + k_2\)
- \(d = x + k_3\)

where \(k_1, k_2, k_3\) are non-negative integers (since \(a \leq b \leq c \leq d\)).

### Step 3: Substituting into the Equation
Substituting these into the equation gives:
\[ x(x + k_1) + (x + k_2)(x + k_3) = x + (x + k_1) + (x + k_2) + (x + k_3) + 3 \]

Expanding both sides:
\[ x^2 + xk_1 + x^2 + (k_2 + k_3)x + k_2k_3 = 4x + k_1 + k_2 + k_3 + 3 \]
Combining like terms:
\[ 2x^2 + (k_1 + k_2 + k_3 + 1)x + k_2k_3 = 4x + k_1 + k_2 + k_3 + 3 \]

### Step 4: Simplifying the Equation
Rearranging gives:
\[ 2x^2 + (k_1 + k_2 + k_3 - 3)x + (k_2k_3 - k_1 - k_2 - k_3 - 3) = 0 \]

### Step 5: Analyzing the Quadratic Equation
For \(x\) to be a positive integer, the discriminant of this quadratic must be a perfect square:
\[ D = (k_1 + k_2 + k_3 - 3)^2 - 8(k_2k_3 - k_1 - k_2 - k_3 - 3) \]

### Step 6: Finding Integer Solutions
We need to find values of \(k_1, k_2, k_3\) such that \(D\) is a perfect square. This can be done by testing small values of \(k_1, k_2, k_3\) and checking the conditions.

### Step 7: Counting Valid Quadruplets
After determining valid combinations of \(k_1, k_2, k_3\) that yield integer values for \(x\), we can count the number of quadruplets \((a, b, c, d)\) that satisfy the original conditions.

### Conclusion
The final answer will be the count of all such valid quadruplets.

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