Let $a<b<c$ be primes such that $c-a$, $c-b$, and $b-a$ are also prime. It is rather simple to show that $(2,5,7)$ is the only triple that satisfies these conditions:
The case $a>2$ reduces to a system of equations with no solution after realizing that each difference must be equal to $2$.
The case $a=2$ reduces to the existence of a prime $p$ such that $p+2$ and $p+4$ are also prime. The only such tuple is $p=3\rightarrow (3,5,7)$. A modular argument takes care of uniqueness.
The tuple $(a,b,c)=(2,5,7)$ follows from $p=3$.
I was wondering if there is a more elegant approach utilizing many number theoretic tools (i.e. elliptic curves, algebraic number theory, ect.)? I realize that this is entirely unnecessary as we can appeal to the most atomic of theory to solve this problem; however, I don't spend much time with number theory and was looking for some application of "modern techniques".