Find all existing solutions of the equation
x^y- y^x= x*y²-19,
provided x and y are primes.

10   for Tot=2 to 9999
20     for X=1 to Tot-1
30         Y=Tot-X
40         Px=prm(X):Py=prm(Y)
50         Lhs=Px^Py-Py^Px
60         Rhs=Px*Py^2-19
70         if Lhs=Rhs then print Px,Py
80     next
90   next

finds only

x=2, y=3
x=2, y=7

and tested all total ordinalities of primes through 231 (for example the 100th prime and the 131st prime have total of their ordinalities at 231), as overflow occurred during the total ordinality of 232.

Posted by Charlie on 2013-09-11 17:08:30