1.Introduction


We show that x1^k + x2^k + x3^k = y1^k + y2^k + y3^k where k = 1,2,6
has infinitely many integer solutions.

Choudhry[1] has already proved that it has infinitely many integer solutions.

We show it by different way.


2.Theorem
         
     
　　 x1^k + x2^k + x3^k = y1^k + y2^k + y3^k where k = 1,2,6 has infinitely many integer solutions.
    
     
Proof.

(ax+b)^k + (cx+d)^k + (x+e)^k - (ax-b)^k - (cx-d)^k - (x-e)^k........................(1)

Case of k=1: (1) is equals to zero where b+d+e=0.....................................(2)

Case of k=2: (1) is equals to zero where ab+dc+e=0...................................(3)

By solving simultaneous equation (2) and (3), we obtain {c,d} as follows.

c=(ab+e)/(b+e), d=-(b+e).............................................................(4)

Case of k=6: 

Substitute (4) to (1), then we find (1) has quadratic forms.


(1) is equals to zero where (4a^3b^3+4a^3be^2+6ea^3b^2+a^3e^3+3a^2b^3+8e^2a^2b+12a^2b^2e
+2a^2e^3+8ab^2e+2b^3a+12abe^2+3e^3a+b^3+4e^3+4b^2e+6be^2)*x^2+b^5+4ab^5+15b^2e^3a
+20b^3e^2a+6be^4a+14b^4ea+e^5a+14be^4+6b^4e+15b^3e^2+20b^2e^3+4e^5 = 0

We have to find rational numbers a, b, and e such that

(-120b^3e^5-10e^7b-180b^6e^2-80b^7e-240b^5e^3-16b^8-e^8-208b^4e^4-45e^6b^2)a^4
+(-190e^6b^2-120b^7e-600b^5e^3-628b^4e^4-360b^6e^2-16b^8-430b^3e^5-6e^8-50e^7b)a^3
+(-11e^8-660b^5e^3-660b^3e^5-325e^6b^2-11b^8-325b^6e^2-90e^7b-90b^7e-828b^4e^4)a^2
+(-50b^7e-628b^4e^4-6b^8-16e^8-190b^6e^2-360e^6b^2-120e^7b-600b^3e^5-430b^5e^3)a
-80e^7b-180e^6b^2-45b^6e^2-b^8-16e^8-10b^7e-120b^5e^3-208b^4e^4-240b^3e^5 = square.

When {b,e}=(7,-8), transform variables then above quartic curve becomes to as follows.

Quartic elliptic curve: V^2 = -90000U^4+510000U^3-1069275U^2+988650U-342225.
 
Let convert quartic elliptic curve to general weierstrass form and we obtain,

Y^2 + XY = X^3 - X^2 - 10584X - 3491910..............................................(5)

The rank of this curve can be determined by mwrank(Cremona), and we know that rank = 1 and generator is  P=(X,Y)=(1539/4 , 54621/8).

Thus (5) has an infinetly many rational solutions, then (1) has an infinetly many integer solutions




Q.E.D.　
 
　　                  
       
3.Example

(b,e)<100
abs(a)<500


  b   e       a        x1     x2     x3     y1     y2     y3

[ 5,  4,  -29/34] [   124,  -303,   185,  -211,   300,   -83]
[ 7,  1, -73/111] [  -140,  -307,   405,  -371,   -43,   372]
[ 7,  4,-173/234] [   140, -1259,  1245, -1351,  1084,   393]
[ 7, -6, 193/400] [  6805, -3015, -4702, -6033, -1181,  6302]
[ 8, -9, 298/203] [  6781, -5392, -1977,  -523, -6305,  6240]
[ 9,  4,  -17/20] [  -627,  -271,   796,  -699,  -167,   764]
[ 9,  4,  -37/44] [  -809,  -535,  1212, -1115,   -93,  1076]
[10,  1,  -24/53] [   178,  -543,   461,  -562,   271,   387]
[10,  1,-116/357] [   637,  -801,   248,  -753,   728,   109]
[11,  1,-125/391] [  1996, -2505,   785, -2371,  2259,   388]
[11,  2,    -3/8] [   393,  -496,   127,  -432,   479,   -23]
[12,  1, -82/269] [  1945, -2179,   300, -2027,  2124,   -31]
[14,  3, -52/101] [   459, -1571,  1388, -1655,   996,   935]
[16,  1, -50/137] [  1229, -2181,  1408, -2179,  1440,  1195]
[19,  3,-137/281] [  1466, -4919,  4317, -5165,  2759,  3270]
[19,  4, -89/210] [  1180, -1935,  1049, -1803,  1676,   421]
[19,  7,  -33/61] [   536, -1077,   673,  -965,   977,   120]
[21,  4,    -1/2] [   791, -2588,  2259, -2716,  1587,  1591]
[25, 19,-259/313] [  3353, -9644,  6627, -7497,  9452, -1619]
[29, 12,-221/394] [ 20225,-36900, 19777,-30612, 34973, -1259]
[31,-29, 377/451] [  2381, -3512,  1791,  1389, -3448,  2719]
[32, 11,-106/195] [   423,  -997,   724,  -953,   852,   251]
[34, 11,-232/353] [  -461, -7113,  8198, -8587,  3642,  5569]
[47, 24, -97/116] [ -1035, -1181,  2120, -2069,   381,  1592]
[53,  9,-149/409] [  9174,-10853,  1775, -9323, 10785, -1366]
[56,-37,  74/435] [  5699, -5255, -1104, -4885, -1664,  5889]
[67,  1,-111/229] [   300, -3475,  3349, -3519,   401,  3292]
[67,-72, 287/212] [ 82507,-41079,-58480,-49215,-50909, 83072]
[71, 12,  -25/58] [  2423, -4796,  3303, -4748,  3587,  2091]
[79,-70, 293/464] [  4355, -3507, -1688, -2597, -2715,  4472]





4.Reference


[1].Ajai Choudhry:On equal sums of sixth powers,Rocky Mountain Journal of Mathematics. 30(2000)