To solve the equation \( x^2 - y^2 = 1298 \), we can start by factoring the left-hand side. The difference of squares can be factored as follows:
\[
x^2 - y^2 = (x - y)(x + y)
\]
Thus, we can rewrite the equation as:
\[
(x - y)(x + y) = 1298
\]
Next, we need to find the pairs of integers \( (x - y) \) and \( (x + y) \) that multiply to give 1298. To do this, we will first find the factors of 1298.
### Step 1: Factor 1298
To factor 1298, we can start by dividing it by the smallest prime numbers:
- 1298 is even, so we can divide by 2:
\[
1298 \div 2 = 649
\]
Now we need to factor 649. We can check for divisibility by prime numbers:
- 649 is not divisible by 2 (it's odd).
- The sum of the digits of 649 is \( 6 + 4 + 9 = 19 \), which is not divisible by 3.
- 649 does not end in 0 or 5, so it is not divisible by 5.
- Dividing by 7:
\[
649 \div 7 \approx 92.71 \quad \text{(not an integer)}
\]
- Dividing by 11:
\[
649 \div 11 \approx 59.91 \quad \text{(not an integer)}
\]
- Dividing by 17:
\[
649 \div 17 \approx 38.76 \quad \text{(not an integer)}
\]
- Dividing by 19:
\[
649 \div 19 \approx 34.16 \quad \text{(not an integer)}
\]
- Dividing by 13:
\[
649 \div 13 \approx 49.15 \quad \text{(not an integer)}
\]
- Finally, dividing by 17:
\[
649 = 17 \times 37
\]
Thus, the complete factorization of 1298 is:
\[
1298 = 2 \times 17 \times 37
\]
### Step 2: Find Factor Pairs
Now we can find the factor pairs of 1298:
- \( (1, 1298) \)
- \( (2, 649) \)
- \( (17, 76) \)
- \( (37, 35) \)
### Step 3: Set Up Equations
For each factor pair \( (a, b) \), we can set:
\[
x - y = a \quad \text{and} \quad x + y = b
\]
From these two equations, we can solve for \( x \) and \( y \):
\[
x = \frac{(x - y) + (x + y)}{2} = \frac{a + b}{2}
\]
\[
y = \frac{(x + y) - (x - y)}{2} = \frac{b - a}{2}
\]
### Step 4: Check for Integral Solutions
For \( x \) and \( y \) to be integers, both \( a + b \) and \( b - a \) must be even.
Now we can check each factor pair:
1. **For \( (1, 1298) \)**:
- \( a + b = 1 + 1298 = 1299 \) (odd)
- \( b - a = 1298 - 1 = 1297 \) (odd)
2. **For \( (2, 649) \)**:
- \( a + b = 2 + 649 = 651 \) (odd)
- \( b - a = 649 - 2 = 647 \) (odd)
3. **For \( (17, 76) \)**:
- \( a + b = 17 + 76 = 93 \) (odd)
- \( b - a = 76 - 17 = 59 \) (odd)
4. **For \( (37, 35) \)**:
- \( a + b = 37 + 35 = 72 \) (even)
- \( b - a = 35 - 37 = -2 \) (even)
### Conclusion
Only the pair \( (37, 35) \) gives us even sums, which means we can find integer solutions for \( x \) and \( y \).
Calculating:
\[
x = \frac{37 + 35}{2} = 36
\]
\[
y = \frac{35 - 37}{2} = -1
\]
Thus, we have one solution \( (36, -1) \) and its symmetric pairs \( (-36, 1) \).
### Final Count of Integral Pairs
The integral pairs \( (x, y) \) satisfying the equation are:
1. \( (36, -1) \)
2. \( (-36, 1) \)
Thus, there are **2 integral pairs** satisfying the equation \( x^2 - y^2 = 1298 \).
