Number of positive integral solutions $(x,y)$ of $x^2-y^2=12345678$

Question

I have to find out the number of positive integral solutions $(x,y)$ of

$$x^2-y^2=12345678$$

Specifically, if $S$ is the set of all ordered pairs $(x,y)$ then $S$ -

A) is an infinite set

B) is the empty set

C) has exactly one element

D) is a finite set and has at least two elements.

Now, with a calculator capable of doing prime factorization, this is an easy question. However without a calculator, its very difficult to find the prime factors by inspection. What is the most efficient and fast solution to such a question.

This question was asked here (Q no. 4)

Answer 1

Hint: Note that $12345678\equiv2\pmod{4}$.

Answer 2

Hint.

$$ 12345678 = 1\times 2\times 3^2\times 47\times 14593 = \prod_i a_i^{b_i} $$

then checking the feasible solutions for

$$ x+y = \frac{\prod_i a_i^{b_i}}{m_k}\\ x-y = m_k $$

for $m_k$ in all possible combinations between the factors, will give us the solution.