1.Introduction

Farzali Izadi and Mehdi Baghalaghdam([1]) showed that x1^5 + x2^5 + x3^5 = y1^3 + y2^3 + y3^3 has infinitely many integer solutions.

We show a simple parametric solution of x1^5 + x2^5 + x3^5 = y1^3 + y2^3 + y3^3.


2.Theorem
      
 
There is a parametric solution of x1^5 + x2^5 + x3^5 = y1^3 + y2^3 + y3^3,

      x1 = a^3(-2c^30+a^30+3a^20b^10+3a^10b^20+b^30)^3
      x2 = b^3(-2c^30+a^30+3a^20b^10+3a^10b^20+b^30)^3
      x3 = c^3(-2c^30+a^30+3a^20b^10+3a^10b^20+b^30)^3
      y1 = (-2c^30+a^30+3a^20b^10+3a^10b^20+b^30)^4(c^30a^5+3c^30b^5+3c^15a^20+6c^15a^10b^10+3c^15b^20+a^35+3a^25b^10+3a^15b^20+a^5b^30)
      y2 = (-2c^30+a^30+3a^20b^10+3a^10b^20+b^30)^4(3c^30a^5+c^30b^5+3c^15a^20+6c^15a^10b^10+3c^15b^20+b^5a^30+3b^15a^20+3b^25a^10+b^35)
      y3 = -(-2c^30+a^30+3a^20b^10+3a^10b^20+b^30)^4c^5(3a^15c^15+3a^10b^5c^15+2a^30+6a^20b^10+6a^10b^20+3b^10a^5c^15+3b^15c^15+2b^30+2c^30)

    
     a, b, c: arbitrary.


     
Proof.

x1^5 +x2^5 +x3^5 = y1^3 + y2^3 + y3^3.....................................(1)

Set x1=a^3, x2=b^3, x3=c^3, y1=t+a^5, y2=t+b^5, y3=mt+c^5.................(2)

(2+m^3)t^3+(3a^5+3b^5+3c^5m^2)t^2+(3a^10+3b^10+3c^10m)t=0

Then we obtain m = -(a^10+b^10)/(c^10) and t = 3(a^5c^15+b^5c^15+a^20+2a^10b^10+b^20)c^15/(-2c^30+a^30+3a^20b^10+3a^10b^20+b^30).

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

 
Q.E.D.
　

3.Reference

[1]. Farzali Izadi and Mehdi Baghalaghdam, ON THE DIOPHANTINE EQUATION IN THE FORM THAT A SUM OF CUBES EQUALS A SUM OF QUINTICS,
Math. J. Okayama Univ. 61 (2019).