Solving the Equation x2 - y2 2017: An Integer Solution Analysis

In this article, we will explore the number of pairs of integer solutions for the equation x2 - y2 2017. We'll start by factorizing the equation using the difference of squares, then proceed to find the integer solutions by analyzing the factorization of 2017.

Factorizing the Equation

The given equation is x2 - y2 2017. Using the difference of squares, we can rewrite it as:

(x - y)(x y) 2017

Let us denote:

a x - y

b x y

Thus, we have:

ab 2017

Since x and y are integers, both a and b must also be integers. We can express x and y in terms of a and b as follows:

x frac{a b}{2}

y frac{b - a}{2}

For x and y to be integers, both a b and b - a must be even. This implies that both a and b must have the same parity, meaning they are both odd or both even. Since 2017 is odd, both a and b must be odd.

To find the odd factor pairs of 2017, we first check if 2017 is prime. We do this by checking for divisibility by prime numbers up to sqrt{2017} approx 44.9. The prime numbers to check are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43.

2017 is not even, so it is not divisible by 2. The sum of the digits 2 0 1 7 10 is not divisible by 3. It does not end in 0 or 5, so it is not divisible by 5. Dividing 2017 by 7 gives approximately 288.14, which is not an integer.

Continuing this process, we find that 2017 is not divisible by any of these primes up to 44. Thus, 2017 is prime and its only positive factors are 1 and 2017.

The odd factor pairs of 2017 are:

(1, 2017)

(2017, 1)

Finding Integer Solutions

Now that we have the factor pairs, we can find the corresponding pairs of x and y:

For (1, 2017): a 1, b 2017 x frac{1 2017}{2} 1009, y frac{2017 - 1}{2} 1008 Thus, one solution is (1009, 1008). For (2017, 1): a 2017, b 1 x frac{2017 1}{2} 1009, y frac{1 - 2017}{2} -1008 Thus, another solution is (1009, -1008).

Since x and y can also be negative, we also have the pairs:

-1009, -1008 -1009, 1008

Thus, the complete set of integer solutions is:

(1009, 1008)

(1009, -1008)

-1009, 1008

-1009, -1008

In total, there are 4 pairs of integer solutions for the equation x2 - y2 2017.