1.Introduction

We show a parametric solution of a(1)x1^n+a(2)x2^n+...+a(k)xk^n=a(1)y1^n+a(2)y2^n+...+a(k)yk^n.
This problem is reduced to solving the linear simultaneous equation, so we can apply this method for any n.


2.Method
        

a(1)x1^n + a(2)x2^n +...+ a(k)xk^n - (a(1)y1^n + a(2)y2^n +...+ a(k)yk^n)=0..............(1)

Let x1=b(1)t+1, x2=b(2)t+1,..., xk=b(k)t+1
    y1=b(1)t-1, y2=b(2)t-1,..., yk=b(k)t-1...............................................(2)
n and b(k) is arbitrary.

Substitute (2) to (1), we obtain the equation (3).

c(1)t^(n-1) + c(2)t^(n-3) + .... + c(m)t^(n-2m+1)=0......................................(3)
n is even: m=n/2
n is odd : m=(n+1)/2
k=m+1

Find the solution of simultaneous equation {c(1)=0,c(2)=0,...,c(m)=0} for {a(1),a(2),...,a(m)}.

Hence we obtain a parametric solution of (1).

     
3.Results


Case 1: ax1^4 + bx2^4 + cx3^4 = ay1^4 + by2^4 + cy3^4

(k,n,m)=(3,4,2)
x1=pt+1, x2=qt+1, x3=t+1 
y1=pt-1, y2=qt-1, y3=t-1

b = -ap(p^2-1)/(q(-1+q^2))
c = ap(p^2-q^2)/(-1+q^2)
a, p, and q are arbitrary.

Example
(p,q,a,b,c)=(3, -2, 1, 4, 5)
(3t+1)^4 + 4(-2t+1)^4 + 5(t+1)^4 = (3t-1)^4 + 4(-2t-1)^4 + 5(t-1)^4


Case 2: ax1^5 + bx2^5 + cx3^5 + dx4^5 = ay1^5 + by2^5 + cy3^5 + dy4^5

(k,n,m)=(4,5,3)
x1=pt+1, x2=qt+1, x3=rt+1, x4=t+1 
y1=pt-1, y2=qt-1, y3=rt-1, y4=t-1

b = -a(-r^2p^2+r^2+p^4-p^2)/((-1+q^2)(q^2-r^2))
c = a(p^4-p^2-q^2p^2+q^2)/(-r^4+q^2r^2-q^2+r^2)
d = -(-q^2p^2+q^2r^2+p^4-r^2p^2)a/((r^2-1)(-1+q^2))
a, p, q, and r are arbitrary.

Example
(p,q,r,a,b,c,d)=(4, 3, 2, 2, -9, 14, -7)
(x1,x2,x3,x4)=(4t+1, 3t+1, 2t+1, t+1)
(y1,y2,y3,y4)=(4t-1, 3t-1, 2t-1, t-1)


Case 3: ax1^6 + bx2^6 + cx3^6 + dx4^6 = ay1^6 + by2^6 + cy3^6 + dy4^6

(k,n,m)=(4,6,3)
x1=pt+1, x2=qt+1, x3=rt+1, x4=t+1 
y1=pt-1, y2=qt-1, y3=rt-1, y4=t-1

b = -(r^2-r^2p^2-p^2+p^4)ap/(q(-r^2+q^2)(q^2-1))
c = ap(q^2-q^2p^2-p^2+p^4)/(r(-q^2+q^2r^2-r^4+r^2))
d = -pa(-q^2p^2+q^2r^2+p^4-r^2p^2)/((q^2-1)(-1+r^2))
a, p, q, and r are arbitrary.


Example
(p,q,r,a,b,c,d)=(4, 3, 2, 1, -6, 14, -14)
(x1,x2,x3,x4)=(4t+1, 3t+1, 2t+1, t+1)
(y1,y2,y3,y4)=(4t-1, 3t-1, 2t-1, t-1)


Case 4: ax1^7 + bx2^7 + cx3^7 + dx4^7 + ex5^7 = ay1^7 + by2^7 + cy3^7 + dy4^7 + ey5^7

(k,n,m)=(5,7,4)
x1=pt+1, x2=qt+1, x3=rt+1, x4=st+1, x5=t+1 
y1=pt-1, y2=qt-1, y3=rt-1, y4=st-1, y5=t-1

b = -a(-p^4-r^2s^2+r^2p^2+s^2r^2p^2+p^6-s^2p^4+s^2p^2-r^2p^4)/(q^6-q^4+s^2r^2q^2+r^2q^2-r^2q^4-s^2q^4+s^2q^2-r^2s^2)
c = (q^2s^2p^2-s^2q^2+s^2p^2-s^2p^4-q^2p^4+p^6+p^2q^2-p^4)a/((r^2-s^2)(q^2-r^2)(r^2-1))
d = -(-r^2q^2+p^2q^2+q^2r^2p^2-q^2p^4+p^6+r^2p^2-r^2p^4-p^4)a/((q^2-s^2)(-s^4+r^2s^2-r^2+s^2))
e = (q^2r^2p^2+q^2s^2p^2-q^2p^4-s^2r^2q^2+s^2r^2p^2+p^6-r^2p^4-s^2p^4)a/((q^2-1)(s^2-1)(r^2-1))
a, p, q, r,and s are arbitrary.


Example
(p,q,r,s,a,b,c,d,e)=(6, 4, 3, 2, 1,-24, 80, -105, 48)
(x1,x2,x3,x4,x5)=(6t+1, 4t+1, 3t+1, 2t+1, t+1)
(y1,y2,y3,y4,y5)=(6t-1, 4t-1, 3t-1, 2t-1, t-1)


Case 5: ax1^8 + bx2^8 + cx3^8 + dx4^8 + ex5^8 = ay1^8 + by2^8 + cy3^8 + dy4^8 + ey5^8

(k,n,m)=(5,8,4)
x1=pt+1, x2=qt+1, x3=rt+1, x4=st+1, x5=t+1 
y1=pt-1, y2=qt-1, y3=rt-1, y4=st-1, y5=t-1

b = -(-r^2s^2-r^2p^4+r^2p^2+s^2r^2p^2+p^6-s^2p^4-p^4+p^2s^2)ap/((-r^2+q^2)(-1+q^2)(-s^2+q^2)q)
c = (-q^2s^2+s^2q^2p^2-q^2p^4+q^2p^2-s^2p^4+p^2s^2+p^6-p^4)ap/((-s^2+r^2)r(-r^4+q^2r^2+r^2-q^2))
d = -ap(-p^4+r^2p^2+p^6+r^2q^2p^2-q^2r^2-q^2p^4+q^2p^2-r^2p^4)/(s(s^6-s^4+r^2q^2s^2-q^2r^2+q^2s^2-q^2s^4-r^2s^4+r^2s^2))
e = (-r^2q^2s^2+r^2q^2p^2-q^2p^4+s^2q^2p^2-r^2p^4+s^2r^2p^2+p^6-s^2p^4)ap/((s^2-1)(r^2-1)(-1+q^2))
a, p, q, r,and s are arbitrary.


Example
(p,q,r,s,a,b,c,d,e)=(5, 4, 3, 2, 1, -8, 27, -48, 42)
(x1,x2,x3,x4,x5)=(5t+1, 4t+1, 3t+1, 2t+1, t+1)
(y1,y2,y3,y4,y5)=(5t-1, 4t-1, 3t-1, 2t-1, t-1)