(* Check for the theorem of Elkahhar  
These parameters r and q will create a,b,c
and k,l,m as stated in the theorem page    *) 
(*--------------------------------*)
r=27    ;
q=3     ; 
(* With the condition r<>q   *)
(*--------------------------------*)
Q:=2*(r^2+q^2)-1
Z:=(Q^2+32*r^2*q^2)*(r^2+q^2)

a:=Z-3*r^2*Q^2
b:=Z-96*r^2*q^4
c:=Z-3*q^2*(Q^2+32*r^4)

k:=Z-3*q^2*Q^2
l:=Z-96*r^4*q^2
m:=Z-3*r^2*(Q^2+32*q^4)

seventh:=(a^7+b^7+c^7)/(k^7+l^7+m^7)
fifth:=(a^5+b^5+c^5)/(k^5+l^5+m^5)
third:=(a^3+b^3+c^3)/(k^3+l^3+m^3)
prod:=(a*b*c)/(k*l*m)

Print["------------------------"]
Print["a= ",a]
Print["b= ",b]
Print["c= ",c]
Print["--------------"]
Print["k= ",k]
Print["l= ",l]
Print["m= ",m]
Print["------------------------"]
TrueQ[seventh==fifth]
TrueQ[seventh==third]
TrueQ[seventh==prod]
Print["------------------------"]
TrueQ[a+b+c==0]
TrueQ[k+l+m==0]
TrueQ[a^2+b^2+c^2==k^2+l^2+m^2]
TrueQ[a^4+b^4+c^4==k^4+l^4+m^4]
Print["------------------------"]

------------------------
a= -2997536049
b= 1754887122
c= 1242648927
--------------
k= 1701813951
l= 1301390802
m= -3003204753
------------------------
True
True
True
------------------------
True
True
True
True
------------------------


    Source: geocities.com/timeparadox