OK, to find

xyz 

     where  
    1. z is the yth prime  
     (starting with 2). 
    2. y is the number of  
    (decimal) digits of x^z.  
    3. x+y+z is the number of  
     (decimal) digits of Gamma(z).  


ouch, by number of decimal digits, I mean the integer part (floor) of the decimal logarithm. I have been warned that one must subtract one unit from these number of digits.

ok,

assuming y and z moderately large, then

1.  y  approx. z/log z (asymptotic law of prime numbers)
2   y  approx z log x/2.3
3   y+z approx z (log z -1) /2.3 (Stirling)
       or  y + 1.4 z approx z log z /2.3

1 and 3:   y2 +1.4 yz approx z2/2.3,  or y/z approx  0.3
          and  (log z)^2 approx 2.3(1+1.4 z/y) approx 14

and 2.  log x approx  2.3 y/z  approx 0.69   weeeellll
  x must be 2, log z close to 4, z in the 50's, and y close to 15.
  
----------------------------------------------

Thanks to all for kind wishes for 2006!

François Glineur  found again unsuspected solutions:
      x=2, y=18, z=61, and    x=-5, y=2, z=3 !!!.

Bruno Vroman  used the Sirling formula in order to get first approximations,
he found the (2,18,61) solution too.


Francois Warichet found (2,18,61) too, and (-2,16,53) after an exhaustive run.

Tom Koornwinder  retaliated by sending a gigantic polynomial.
.