1.Introduction

Farzali Izadi and Mehdi Baghalaghdam[1] showed the parametric solution of m(x1^k+x2^k+...+xt^k)=n(y1^k+y2^k+...+yt^k),k=1,3.

We show the parametric solution of m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3) different from their one.
This parametric solution gives infinite integral solutions for any m,n.

2.Theorem
        
     

    There is a parametric solution of m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3).
    
    x1 = (3nm^3a^6+6nm^3a^2-6m^2n^2a^4+3n^3ma^2-3m^2a^6n^2-6n^2m^2a^2+6mn^3a^4+3nm^3a^4-3m^4a^4-3m^2n^2-3n^4a^2+3mn^3)p^2
       + (-6mn^3a^4+6m^2an^2+12m^2a^5n^2+6nm^3a^2+6m^4a^5+6mn^3a^6+12n^3ma^2-12m^2n^2a^4-6mn^3-6m^2n^2+6n^4a^2-6n^4a^4-12m^3a^3n-6m^3a^7n-6m^4a^2+6nm^3+6nm^3a^4-6ma^3n^3)p
       - mn^3+m^4a^6-4n^4a^6+4nm^3-9n^2m^2a^2-3nm^3a^4-4m^4+n^4+6nm^3a^2+3m^2n^2a^4-2mn^3a^6-15n^3ma^2-nm^3a^6+12mn^3a^4+6m^4a^3+6n^4a^4+6m^2n^2+12m^2a^3n^2-12m^3a^5n-6ma^5n^3+6m^2a^7n^2-6m^3an
       
    x2 = (-3nm^3a^6-6nm^3a^2+6m^2n^2a^4-3n^3ma^2+3m^2a^6n^2+6n^2m^2a^2-6mn^3a^4-3nm^3a^4+3m^4a^4+3m^2n^2+3n^4a^2-3mn^3)p^2
       + (-18mn^3a^4+6m^4a^5-6nm^3a^2+6m^2an^2+12m^2a^5n^2+12m^2n^2a^4-6mn^3-6mn^3a^6-12n^3ma^2-12m^3a^3n+12m^2a^6n^2+6m^2n^2+6n^4a^2+6n^4a^4-6ma^3n^3-6m^3a^7n+6m^4a^2-6nm^3+24n^2m^2a^2-18nm^3a^4)p
       + 6m^2n^2-18nm^3a^2+12m^2a^3n^2-12m^3a^5n-6ma^5n^3+6m^2a^7n^2-4nm^3+9n^2m^2a^2-6m^3an+4m^4-n^4-12mn^3a^4+6m^4a^3+6n^4a^4+4n^4a^6+nm^3a^6+mn^3-m^4a^6-10mn^3a^6-9n^3ma^2+3nm^3a^4+21m^2n^2a^4
       
    x3 = (6m^2a^6n^2+12n^2m^2a^2-12mn^3a^4-6nm^3a^4+6n^4a^2-6mn^3+3amn^3-3n^4a^3+6ma^5n^3+3m^3a^5n-3m^2a^7n^2-6m^2a^3n^2-3ma^3n^3-6m^3a^3n+6m^2a^5n^2-3m^3a^7n+3m^2an^2+3m^4a^5)p^2
       + (-12m^2a^5n^2-6n^4a^5-6mn^3a^6-12n^3ma^2+6n^4a^4+12m^2n^2a^4-6nm^3a^2+6mn^3a^7+12ma^3n^3+6m^3a^3n-6m^2an^2+6m^2n^2)p
       - 9m^2a^5n^2-m^3a^7n-3ma^3n^3+12m^3a^3n-3m^3a^5n+12ma^5n^3+4m^3an-4n^4a^7-amn^3+4mn^3a^7+m^4a^7-9m^2a^3n^2-4am^4+an^4
       
    x4 = (-3m^4a^5+3m^3a^7n-3m^3a^5n+3m^2a^7n^2+3n^4a^3-6m^2a^5n^2-6ma^5n^3+6m^3a^3n-6m^2a^6n^2-12n^2m^2a^2+12mn^3a^4+6nm^3a^4-6n^4a^2+6mn^3+3ma^3n^3+6m^2a^3n^2-3m^2an^2-3amn^3)p^2
       + (-12m^2a^7n^2-24m^2a^3n^2-6mn^3a^7+18nm^3a^2-12ma^3n^3-6m^3a^3n-18m^2n^2-36m^2n^2a^4-18n^4a^4+6n^4a^5+12ma^5n^3+24m^3a^5n+6m^2an^2-12m^4a^3+18mn^3a^6+36n^3ma^2+12m^2a^5n^2+12m^3an)p
       - 12n^4a^6+12nm^3-36n^2m^2a^2+m^3a^7n+3ma^3n^3+24m^3a^3n-27m^2a^5n^2-8am^4-an^4+9m^2a^3n^2+4n^4a^7-12ma^5n^3+36mn^3a^4+3m^3a^5n+8mn^3a^7-4m^3an+amn^3-m^4a^7
       
    y1 = (-3m^4a^5+3m^3a^7n+3m^3a^5n-3m^2a^7n^2-3n^4a^3-6m^2a^5n^2+6ma^5n^3+6m^3a^3n+3ma^3n^3-6m^2a^3n^2-3m^2an^2+3amn^3)p^2
       + (6ma^3n^3+6mn^3a^7+6m^4a^5-6m^3a^3n-12mn^3a^4+6ma^5n^3-6m^4a^3+6m^3an-12m^2a^3n^2-6nm^3a^4+6m^2a^6n^2-6n^4a^5+12n^2m^2a^2-6mn^3+6n^4a^2-6m^3a^7n+12m^3a^5n-6m^2a^7n^2)p
       + 6m^4a^3+6n^4a^4+6m^2n^2-m^3a^7n+3m^2a^3n^2-amn^3-9m^2a^5n^2-6mn^3a^6-3ma^3n^3+12m^3a^3n+an^4+6ma^5n^3-12n^3ma^2+m^4a^7-15m^3a^5n+12m^2n^2a^4+6m^2a^7n^2+4mn^3a^7-6nm^3a^2-4n^4a^7-2m^3an-4am^4
       
    y2 = (3m^4a^5-3m^3a^7n-3m^3a^5n+3m^2a^7n^2+3n^4a^3+6m^2a^5n^2-6ma^5n^3-6m^3a^3n-3ma^3n^3+6m^2a^3n^2+3m^2an^2-3amn^3)p^2
       + (-18ma^3n^3+6m^2a^7n^2-6mn^3a^7+12m^2a^3n^2+12m^2an^2-18m^3a^3n+6m^4a^5+12n^2m^2a^2-12mn^3a^4-6nm^3a^4-6ma^5n^3+24m^2a^5n^2-6m^3a^7n+6n^4a^5+6m^2a^6n^2-6m^3an-6mn^3+6n^4a^2-12m^3a^5n+6m^4a^3)p
       + 6m^4a^3+6n^4a^4+6m^2n^2+m^3a^7n+21m^2a^3n^2-9m^3a^5n+9m^2a^5n^2+4am^4+3ma^3n^3-12m^3a^3n+4n^4a^7-18ma^5n^3-an^4+amn^3-6nm^3a^2+12m^2n^2a^4+6m^2a^7n^2-12n^3ma^2-4mn^3a^7-6mn^3a^6-m^4a^7-10m^3an
       
    y3 = (6m^4a^5-6m^3a^7n+12m^2a^5n^2-12m^3a^3n+3nm^3a^6+6nm^3a^2-6m^2n^2a^4+3n^3ma^2+3m^2a^6n^2+6n^2m^2a^2-6mn^3a^4-3nm^3a^4-3m^4a^4-3m^2n^2+3n^4a^2-3mn^3-6ma^3n^3+6m^2an^2)p^2
       + (6m^2a^7n^2-6ma^5n^3-12m^3a^5n+6m^4a^3+12m^2a^3n^2+6mn^3a^4-6m^4a^2+6nm^3-6m^3an-12n^2m^2a^2+12nm^3a^4-6m^2a^6n^2)p
       + m^4a^6+12nm^3a^2+4nm^3-mn^3-3nm^3a^4-4n^4a^6+n^4-9n^2m^2a^2-9m^2n^2a^4-4m^4-3n^3ma^2+12mn^3a^4+4mn^3a^6-nm^3a^6
       
    y4 = (6m^3a^7n+3nm^3a^4+6m^2n^2a^4-3m^2a^6n^2-12m^2a^5n^2-3n^3ma^2+3m^4a^4+12m^3a^3n-3nm^3a^6+6mn^3a^4+3m^2n^2-3n^4a^2+3mn^3-6nm^3a^2-6m^4a^5+6ma^3n^3-6m^2an^2-6n^2m^2a^2)p^2
       + (12mn^3a^6+24n^3ma^2+18ma^5n^3+12nm^3a^2+36m^3a^5n-18m^4a^3+18m^3an-18m^2a^7n^2+6m^4a^2-6nm^3+12n^2m^2a^2-24m^2n^2a^4-6mn^3a^4+6m^2a^6n^2-12n^4a^4-36m^2a^3n^2-12nm^3a^4-12m^2n^2)p
       - 4mn^3a^6+3n^3ma^2+nm^3a^6-36m^2a^5n^2+36m^3a^3n+3nm^3a^4+4m^4+24mn^3a^4+12mn^3a^7-12am^4-m^4a^6-12nm^3a^2+8nm^3-27n^2m^2a^2+mn^3-8n^4a^6-n^4+9m^2n^2a^4
       
    m,n,a,p are arbitrary.


 
Proof.

m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3).................................(1)

Let x1=t+1, x2=t-1, x3=pt+a, x4=qt-a, y1=t+a, y2=t-a, y3=pt+1, y4=qt-1........(2)

Substitute (2) to (1) and simplifying (1), we obtain

(2m+mq^3-2n-nq^3-np^3+mp^3)t^3+(-3mq^2a+3mp^2a+3nq^2-3np^2)t^2
+(3mpa^2+6m-3nq+3mqa^2-3np-6na^2)t=0..........................................(3)

Equating to zero the coefficient of t, then we obtain

q = -(mpa^2+2m-np-2na^2)/(-n+ma^2)

Finally, we obtain t as follows

t = -6(m^3pa^4n-m^2pa^6n^2-2m^2n^2pa^2+2mn^3pa^4+nm^3a^2-2m^2n^2a^4+mn^3a^6+2n^3ma^2-m^4a^3-n^4a^4-m^2n^2-m^4a^5p-2m^2a^3n^2+2m^3a^5n+ma^5n^3-m^2a^7n^2+mn^3p-n^4pa^2+m^3an-2m^2a^5pn^2+m^3a^7pn-m^2an^2p+ma^3n^3p+2m^3a^3pn)
  /(3m^3p^2a^6n+6m^3p^2a^2n+12m^3pa^4n-6m^2p^2a^4n^2-6m^2pa^6n^2-12m^2n^2pa^2+3mn^3p^2a^2+6mn^3pa^4-3m^2p^2a^6n^2-6m^2p^2a^2n^2-12m^2pa^4n^2+6mp^2a^4n^3+6mpa^6n^3+12mn^3pa^2+3nm^3p^2a^4+6nm^3pa^2-mn^3+m^4a^6-4n^4a^6+4nm^3
   -9n^2m^2a^2-3nm^3a^4-4m^4+n^4-3m^4p^2a^4-6m^4pa^2+6m^3np+12nm^3a^2-3m^2n^2p^2-9m^2n^2a^4+4mn^3a^6-3n^3ma^2-nm^3a^6-6m^2n^2p-3n^4p^2a^2-6n^4pa^4+3mn^3p^2+12mn^3a^4)

Substitute t to (2), and obtain a parametric solution.            


Q.E.D.　
 
　　                  
       
3.Examples

a=2

x1 = (-4n^4+88nm^3-16m^4-105m^2n^2+37mn^3)p^2+(-24n^4+94mn^3+66m^2n^2+56m^4-246nm^3)p-160nm^3+36m^4-63mn^3+294m^2n^2-53n^4

x2 = (4n^4-88nm^3+16m^4+105m^2n^2-37mn^3)p^2+(40n^4-258mn^3+486m^2n^2+72m^4-394nm^3)p-120nm^3-4m^4-353mn^3+414m^2n^2+117n^4

x3 = (-144nm^3+32m^4+66m^2n^2-8mn^3)p^2+(144mn^3-66m^2n^2-32n^4+8nm^3)p+40m^4+290mn^3-40nm^3-170n^4-120m^2n^2

x4 = -2*(-n+4m)*(4m^3-17nm^2+4mn^2)p^2-2*(-n+4m)*(72mn^2-33nm^2-16n^3+4m^3)p-2*(-n+4m)*(6m^3-16nm^2+35mn^2-43n^3)

y1 = (-8n^4+176nm^3-32m^4-210m^2n^2+74mn^3)p^2+(-56n^4+270mn^3-144m^2n^2+48m^4-172nm^3)p-180nm^3+56m^4+82mn^3+234m^2n^2-138n^4

y2 = (8n^4-176nm^3+32m^4+210m^2n^2-74mn^3)p^2+(72n^4-434mn^3+696m^2n^2+80m^4-468nm^3)p-100nm^3-24m^4-498mn^3+474m^2n^2+202n^4

y3 = (4n^4+48m^4-232nm^3-45mn^3+171m^2n^2)p^2+(-32mn^3+144m^2n^2+8m^4-66nm^3)p-20nm^3+20m^4+145mn^3-60m^2n^2-85n^4

y4 = (-4n^4+232nm^3-48m^4-171m^2n^2+45mn^3)p^2+(-852m^2n^2+346nm^3-40m^4+448mn^3-64n^4)p-171n^4+120nm^3-28m^4-372m^2n^2+559mn^3

(m,n,a)=(3,2,2)

(x1,x2,x3,x4) = ( p^2-9p+5, -p^2-7p+3, -6p^2+2p+2, 6p^2-10p+2 )
(y1,y2,y3,y4) = ( 2p^2-10p+6, -2p^2-6p+2, -7p^2+3p+1, 7p^2-11p+3 )

In this way, this parametric solution gives infinite solutions.




 

4.Reference
[1]: Farzali Izadi and Mehdi Baghalaghdam, On the simultaneous Diophantine equations
     m(x1^k+x2^k+...+xt^k)=n(y1^k+y2^k+...+yt^k),k=1,3, https://arxiv.org/pdf/1705.01381.pdf