1.Introduction

We show equation ax^n + by^n + cz^n + dw^n = 0 has a simple parametric solution.


2.Theorem
        
     
ax^n + by^n + (p^nr-a)z^n + (-q^nr-b)w^n = 0 has a simple parametric solution.

(x,y,z,w)=(q, p, q, p)
a,b,n,p,q,r are arbitrary.

 
Proof.

ax^n + by^n + cz^n + dw^n = 0.........................................(1)

Let assume equation(1) has a solution (x,y,z,w)=(q, p, q, p).

Substitute c = p^nr-a and d = -q^nr-b, and simplifying (1), then we obtain a parametric solution as follows.

ax^n + by^n + (p^nr-a)z^n + (-q^nr-b)w^n = 0..............................(2)
   
(x,y,z,w) = (q, p, q, p)
a,b,n,p,q,r are arbitrary.


Examples:

a=17,b=13,n=3: 17x^3 + 13y^3 + (p^3r-17)z^3 + (-q^3r-13)w^3 = 0

a=19,b=17,n=4: 19x^4 + 17y^4 + (p^4r-19)z^4 + (-q^4r-17)w^4 = 0

a=23,b=19,n=5: 23x^5 + 19y^5 + (p^5r-23)z^5 + (-q^5r-19)w^5 = 0



Q.E.D.