Yet again a conjecture!
Motivated by Catalan's conjecture and a recent question of mine, I conjecture that
For distinct, positive integers $a,b$, the only solution to this equation $$a^b-b^a=a+b\tag1$$ is $(a,b)=(2,5).$
It is of anticipation that there will be much fewer solutions for increasing $a,b$ since the values that $a^b-b^a$ are extremely restricted due to the small growth rate of $a+b$.
Plotting $x^y-y^x=x+y$ on Desmos strongly suggests that are no other solutions other than those on the line $y=x$.
Is there something obvious I'm missing to solve this?
