Diophantine Equation Help

Question

Exam question stumped me, can I have some help?

Find all the positive integer solutions of to: $35=x^2-y^2$

Answer 1

hint

$$x^2-y^2=(x-y)(x+y) $$ $$35=1\times 35=5\times 7$$ thus $$x+y=35 ; x-y=1$$ or $$x+y=7 ; x-y=5$$ finally, the solutions set is $$S=\{(18,17),(6,1)\} $$

Answer 2

This is similar to the $\,A$-function of Eucid's formula $\,F(m,k)\,$ for generating Pythagorean triples, shown here as $\,A=m^2-k^2.\quad$ We can find the $\,(m,k)\,$ values by solving for $\,k\,$ and testing a defined range of $\,m$-values to see which yield integers. $$A=m^2-k^2\implies k=\sqrt{m^2-A}\quad\text{for}\quad \sqrt{A+1} \le m \le \frac{A+1}{2}$$ The lower limit ensures $k\in\mathbb{N}$ and the upper limit ensures $m> k$. $$A=35\implies \sqrt{35+1}=6\le m \le \frac{35+1}{2} =18\\ \text{and we find}\quad m\in\{6,18\}\implies k \in\{1,7\} $$ Using all three functions of Euclid's formula yields the following triples. $$F(6,1)=(35,12,37)\qquad F(18,17)=(35,612,613)\qquad $$