<s>[INST] Existence of two primes satisfying the given conditions I want to know whether 
the equation 
$x^a-x=y^b-y$
has a solution or not satisfying the conditions that $x$ and $y$ are distinct odd primes, $a$ and $b$ are integers both greater than $1$.
 [/INST] Michael Bennett, On some exponential equations of S. S. Pillai, Canadian Journal of Math. 53 (2001) 897-922, reports eight solutions of $x^a-x=y^b-y$ in positive integers, with $a,b>1$:  
$6=2^3-2=3^2-3$,
$30=2^5-2=6^2-6$,
$210=6^3-6=15^2-15$,
$240=3^5-3=16^2-16$,
$2184=3^7-3=13^3-13$,
$8190=2^{13}-2=91^2-91$,
$78120=5^7-5=280^2-280$,
$24299970=30^5-30=4930^2-4930$. 
See the OEIS for links. I don't think any new ones have been discovered, and I don't think anyone has proved there are none. Of those eight solutions, the only one satisfying the (rather unmotivated) primality stipulations is the one Barry reported in his answer five years ago. 
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