1.Introduction


A. Bremner and J. Delorme showed that x1^k+x2^k+x3^k+x4^k+x5^k+x6^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k   
for k=1,2,3,9 has infinetly many integer solutions.([1])

By differrent way,Tito Piezas([2]) showed that above equation has infinetly many solutions.

A.Choudhry showed that x1^k+x2^k+x3^k = y1^k+y2^k+y3^k,for k=1,2,6
has infinetly many integer solutions.([3])

I will show the infinity of the solution for above (k.6.6) equation.

My method is depends on their method.

I think it interesting that Choudhry's parameter of sixth powers is effective for ninth powers.


2.x1^k+x2^k+x3^k+x4^k+x5^k+x6^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k for k=1,2,3,9
  has infinetly many solutions.
         
     
     
Proof.

x1^k+x2^k+x3^k+x4^k+x5^k+x6^k = y1^k+y2^k+y3^k+y4^k+y5^k+y6^k................(1)

Set variables as following,

    x1 = 2(a+b)m+(a-b+t)n+w      y1 = 2(a+b)m+(a-b+t)n-w
    x2 = -2am+(a+b+t)n+w         y2 = -2am+(a+b+t)n-w
    x3 = -2bm-(a+b-t)n+w         y3 = -2bm-(a+b-t)n-w
    x4 = -2(a+b)m+(a-b+t)n-w     y4 = -2(a+b)m+(a-b+t)n+w
    x5 = 2am+(a+b+t)n-w          y5 = 2am+(a+b+t)n+w
    x6 = 2bm-(a+b-t)n-w          y6 = 2bm-(a+b-t)n+w.........................(2)


By using Choudhry's parameter([3]),(1) is always equals to zero for k=1,2,3.


Case of k=9: 
x1^9+x2^9+x3^9+x4^9+x5^9+x6^9 - ( y1^9+y2^9+y3^9+y4^9+y5^9+y6^9)
=18432abmn^2(m-n)(m+n)(a+b)(a-b+3t)
*(-5m^2b^3-n^2b^3+7m^2b^2t-4m^2b^2a+7n^2b^2t+2n^2b^2a+7am^2tb+4m^2ba^2-14n^2bt^2-2n^2ba^2-7an^2bt+5m^2a^3+7m^2a^2t+7n^2a^2t+14n^2at^2+n^2a^3+14n^2t^3)
*(2m^2b^2+2m^2ba+2m^2a^2+3n^2a^2+14n^2ta+21n^2t^2-4bn^2a-14bn^2t+3b^2n^2)

So,we have to find the rational solution (m,n) of 

(-14bt^2-7bat+a^3+2ab^2+14at^2-2a^2b-b^3+7a^2t+7b^2t+14t^3)n^2+(-4ab^2-5b^3+4a^2b+5a^3+7b^2t+7a^2t+7bat)m^2=0.
..............................................................................(3)

So that there are rational solution,-(-14bt^2-7bat+a^3+2ab^2+14at^2-2a^2b-b^3+7a^2t+7b^2t+14t^3)*(-4ab^2-5b^3+4a^2b+5a^3+7b^2t+7a^2t+7bat)

must be square number,then we have to find rational solution (a,b,t,s) of

s^2 = (-98a^2-98ab-98b^2)t^4+(56ab^2-56a^2b+168b^3-168a^3)t^3
    + (63a^2b^2-119a^4+14ab^3-119b^4+14a^3b)t^2
    + (14a^3b^2-14a^2b^3-42a^5+42b^5-14ab^4+14a^4b)t
    + 6ab^5+6a^5b-5a^6+2a^4b^2-5b^6-6a^3b^3+2a^2b^4............................(4)

By brute force method,we can find a solution (a,b,t)=(3, 4, 27/41).

Substitute (a,b,t)=(3, 4, 27/41) to (3),then (3) becomes to 160/68921(139n-164m)(139n+164m)=0.

So,we get (m,n)=(139,164).

Substitute (a,b,t,m,n)=(3, 4, 27/41,139,164) to (2),then we get follwing solution

[x1, x2, x3, x4, x5, x6] = [1025,  291,   -996,  -1081,   965,    -44]
[y1, y2, y3, y4, y5, y6] = [ 865,  131,  -1156,   -921,  1125,    116].
(common factors were removed)



Next,substitute (a,b)=(3,4) to (4),then we get a quartic equation.

s^2 = -3626*t^4+6888*t^3-26831*t^2+24570*t-3029................................(5)

Transform (5) to minimal Weierstrass form (6).

V^2 + UV + V = U^3 -7166374 -22875861928.......................................(6)

We get a point P(U,V) = (1026337/64, -1026359837/512).

As this point on the curve (6) does not have intger coordinates,
there are infinitely many rational points on the curve (6) by Nagell-Lutz theorem.

By using point 2P(t,s)=(3181201/12876603, 6408411316637440/165806904819609),
we obtain a new solution.


[x1, x2, x3, x4, x5, x6] =[15677071397,40208111671,-63297775068,-26458358421,63560861593,-33396207172]
[y1, y2, y3, y4, y5, y6] =[19383367397,43914407671,-59591479068,-30164654421,59854565593,-37102503172]





Q.E.D.　
 
　　                  
       
3.Small Solutions

(a,b,height(t))<50




[ a,  b,      t]  [   x1     x2      x3     x4     x5       x6]  [   y1     y2     y3      y4     y5       y6]  


[1,   3,    6/5], [    18,    13,    -15,   -22,    13,      1], [   10,     5,    -23,    -14,    21,      9]
[3,   4,  27/41], [  1025,   291,   -996, -1081,   965,    -44], [  865,   131,  -1156,   -921,  1125,    116]
[4,   9,   13/3], [   453,   -98,   -307,  -455,   150,    281], [  429,  -122,   -331,   -431,   174,    305]
[4,   9,  29/11], [  1217,  1015,  -1368, -1568,  1307,   -171], [  785,   583,  -1800,  -1136,  1739,    261]
[4,  45,   25/3], [-12318, 11233, -16435, -5567, 20157,  -5830], [-3558, 19993,  -7675, -14327, 11397, -14590]
[7,   9,  30/29], [  2265,  1252,  -2973, -2503,  2947,   -716], [ 1993,   980,  -3245,  -2231,  3219,   -444]
[11, 16,  29/35], [ -1395, 19149, -26026, -5467, 26629, -17026], [ 2741, 23285, -21890,  -9603, 22493, -21162]
[11, 41,   38/7], [ -6609,  8441, -12680, -3109, 14272,  -5739], [-1185, 13865,  -7256,  -8533,  8848, -11163]
[12, 47,   19/3], [ -4758,  5707,  -8485, -1993,  9679,  -3918], [ -990,  9475,  -4717,  -5761,  5911,  -7686]
[13, 21,  24/17], [ -1557, 16717, -23928, -6115, 24520, -14021], [ 2827, 21101, -19544, -10499, 20136, -18405]
[13, 36,   27/7], [ -3881,  5286,  -7245, -1010,  8219,  -4289], [ -961,  8206,  -4325,  -3930,  5299,  -7209]
[16, 21,  47/17], [  1211,   586,  -1461, -1325,  1442,   -285], [ 1043,   418,  -1629,  -1157,  1610,   -117]
[31, 37,   26/5], [   453,   -98,   -307,  -455,   281,    150], [  429,  -122,   -331,   -431,   305,    174]
[32, 39,   37/9], [ 21798,  7973, -25115,-23059, 24813,  -4082], [19470,  5645,  -27443,-20731, 27141,  -1754]




4.References

[1].A. Bremner and J. Delorme: ON EQUAL SUMS OF NINTH POWERS, Math. Comp.79 ,2010

[2].Tito Piezas:http://sites.google.com/site/tpiezas/029

[3].A.Choudhry: ON EQUAL SUMS OF SIXTH POWERS, Rocky Mountain Journal of Mathematics,V30,2000