Prove that the Diophantine equation has only one solution for $a,b,c$. Prove that the Diophantine equation
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$     
has only one solution for positive integers $a,b,c$   with $a<b<c$

there must be a number smaller than $3$ since there is going to be a number larger than $3$. So $a=2$. we now want $\frac{1}{b}+\frac{1}{c}=\frac{1}{2}$ with $b$ and $c$ larger than $2$. If we take $\frac{1}{4}$ we need to repeat it, so we need a number larger than $\frac{1}{4}$, this forces us to take $\frac{1}{3}$.
