Since x and y are prime and we have powers, it makes sense to evaluate the equation mod x and mod y, with help from Fermat's Little Theorem.
Working mod x we get -y = -19 mod x, which means x divides (y-19)
Working mod y we get x = 19 mod y, which means y divides (x+19)
Now multiply the two divisibility conditions together: x*y divides (y-19)*(x+19)=x*y-19x+19y-19^2
Simplifying a bit we can say x*y divides 19*(x-y+19).
So one set of cases is apparent, one of x or y is 19. In either case the right side of the equation is a multiple of 19 and one of the two terms on the left is a multiple of 19. But then working mod 19 this means that the other term is a multiple of 19, which then means both x and y equal 19.
This is a problem since then the left side reduces to 0 but the right side does not. So no solutions so far.
So neither x nor y is 19 then x*y divides 19*(x-y+19) can be reduced to x*y divides (x-y+19).
So now I'll introduce an inequality. x*y divides (x-y+19) implies x*y <= |x-y+19|. We can drop the absolute values by noting for positive x and y we must have |x-y+19| < x+y+19.
Then x*y < x+y+19.
Rearrange this into (x-1)*(y-1) < 20. With x and y being primes then x-1 and y-1 can only be from the set {1,2,4,6,10,12,16,18}
This leaves a finite set of possible solutions, which Charlie's exploration has already covered giving the two known solutions of (x,y) = (2,3) or (2,7)
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