Thank you for pointing out this trick. The following methods are equivalent:
In[9]:= FindInstance[ x^4 + y^4 + z^4 == w^4 &&x>0&&y>0&&z>0, {x,y,z,w},Integers]
FindInstance[ x^4 + y^4 + z^4 == w^4, {x,y,z,w},PositiveIntegers]
Out[9]= {{x -> 18796760, y -> 2682440, z -> 15365639, w -> 20615673}}
Out[10]= {{x -> 18796760, y -> 2682440, z -> 15365639, w -> 20615673}}
Furthermore, I also noticed that in order to obtain the same result as the Wiki entry, at least two variables need to be set to known values:
In[33]:= FindInstance[ {x^4 + y^4 + z^4 == w^4, x==95800 }, {x,y,z,w},PositiveIntegers]
FindInstance[ {x^4 + y^4 + z^4 == w^4, x==95800,y==217519 }, {x,y,z,w},PositiveIntegers]
During evaluation of In[33]:= FindInstance::nsmet: The methods available to FindInstance are insufficient to find the requested instances or prove they do not exist.
Out[33]= FindInstance[{x^4 + y^4 + z^4 == w^4, x == 95800}, {x, y, z,
w}, PositiveIntegers]
Out[34]= {{x -> 95800, y -> 217519, z -> 414560, w -> 422481}}
Nevertheless, the following method still fails to do the trick:
In[44]:= FindInstance[ {x^4 + y^4 + z^4 == w^4, x==95800, w == 422481}, {x,y,z,w},PositiveIntegers]
Out[44]= $Aborted
