Solutions to $p_1 + p_2^{a} = p_2 + p_1^{b}$ where $p_1,p_2$ are prime and $a,b$ are positive integers

Question

Solutions to $p_1 + p_2^{a} = p_2 + p_1^{b}$ where $p_1,p_2$ are prime and $a,b$ are positive integers.

Trivial cases are when $a=b=1$. Are there any other solutions?

Answer 1

HINT

\begin{aligned} p_1 + p_2 ^ a = p_2 + p_1 ^ b &\iff \\ p_2 ^ a - p_2 = p_1 ^ b - p_1 &\iff \\ p_2 ( p_2 ^ {a-1} - 1) = p_1 ( p_1 ^ {b-1} - 1) \end{aligned}

since $p_2$ and $( p_2 ^ {a-1} - 1)$ are co-primes (as well as $p_1$ and $( p_1 ^ {b-1} - 1)$) it must hold that either

• $p_2 = p_1 \Rightarrow a = b$

or

• $p_2 = p_1 ^{b-1} -1 \Rightarrow \ldots$