In[]:=
FindInstance1+
1
x
1+
1
y
1+
1
z
==3,{x,y,z},PositiveIntegers,3
FindInstance
:The methods available to FindInstance are insufficient to find the requested instances or prove they do not exist.
Out[]=
FindInstance1+
1
x
1+
1
y
1+
1
z
3,{x,y,z},

>0
,3
In[]:=
FindInstance1+
1
x
1+
1
y
1+
1
z
==3,{x,y,z},PositiveIntegers
Out[]=
{{x2,y2,z3}}
In[]:=
FindInstance1+
1
x
1+
1
y
1+
1
z
==3,x!=2,y!=2,z!=2,{x,y,z},PositiveIntegers
Out[]=
{{x8,y3,z1}}
In[]:=
FindInstance1+
1
x
1+
1
y
1+
1
z
==3,x!=2,y!=2,z!=2,x!=3,y!=3,z!=3,{x,y,z},PositiveIntegers
Out[]=
{{x1,y4,z5}}