1.Introduction


By Tito Piezas[1],it seems that x1^k + x2^k + … + x6^k = y1^k + y2^k + … + y6^k , for k = 2,4,6,8
has a follwing parametric solution.

(pa+qb+c)^k+ (pa+qb-c)^k+ (qa-pb+d)^k+ (qa-pb-d)^k+ (ra+sb)^k+ (sa-rb)^k=

(pa-qb+c)^k+ (pa-qb-c)^k+ (qa+pb+d)^k+ (qa+pb-d)^k+ (ra-sb)^k+ (sa+rb)^k,
for k = 2,4,6,8 where ma^2+nb^2 = c^2 and na^2+mb^2 = d^2. 


By using above general form,I found a parametric solution of x1^k + x2^k + … + x8^k = y1^k + y2^k + … + y8^k,
for k = 2,4,6,8.




[1].Tito Piezas:http://sites.google.com/site/tpiezas/028





2.Theorem
         
     
　　(2a+b+c)^k +(2a+b-c)^k +(a-2b+d)^k +(a-2b-d)^k +(3a+2b+e)^k +(3a+2b-e)^k +(2a-3b+e)^k +(2a-3b-e)^k

   =(2a-b+c)^k +(2a-b-c)^k +(a+2b+d)^k +(a+2b-d)^k +(3a-2b+e)^k +(3a-2b-e)^k +(2a+3b+e)^k +(2a+3b-e)^k
   for k = 2,4,6,8 has infinetly many solutions where a^2-5b^2+c^2=5e^2 and 6a^2-6b^2+c^2=d^2.

     
Proof.

 (pa+qb+c)^k+(pa+qb-c)^k+(qa-pb+d)^k+(qa-pb-d)^k+(ra+sb+e)^k+(ra+sb-e)^k+(sa-rb+e)^k+(sa-rb-e)^k
-(pa-qb+c)^k-(pa-qb-c)^k-(qa+pb+d)^k-(qa+pb-d)^k-(ra-sb+e)^k-(ra-sb-e)^k-(sa+rb+e)^k-(sa+rb-e)^k
....................................................(1)

When (p,q,r,s)=(2,1,3,2),above equation has a non trivial solution.


Case of k=2: (1) is always equals to zero.

Case of k=4: (1) is equals to zero where 6a^2-6b^2+c^2=d^2.   

Case of k=6 and k=8: (1) is equals to zero where a^2-5b^2+c^2=5e^2.  


Consequently,we can obtain a solution by solving simultaneous equation {a^2-5b^2+c^2=5e^2,6a^2-6b^2+c^2=d^2}.

These two quadratic equations define an elliptic curve E:V^2 = 75625*U^4-189000*U^3+139600*U^2-30240*U+1936.

Let convert quartic curve to general Weierstrass form  Y^2 = X^3 + X^2 - 136633*X + 19329863.

The rank of this curve can be determined by mwrank(Cremona),and we know that rank = 2.

The generator1 is  P=(X,Y)=(323 , -3000).
The generator2 is  Q=(X,Y)=(707 , -16632).

We obtain 2P=(X,Y)=(893/4, 459/8) <======> (U,V)=(-3816/1075, -218435444/46225).

Thus (1) has an infinetly many solutions.




Q.E.D.　
 
　　                  
       
3.Example

(a,b,c)<100

terms: 0=k.8.8, 1=k.7.7


[ a, b, c, d,    e][ x1  x2  x3  x4  x5  x6  x7  x8] [ y1  y2  y3  y4  y5  y6  y7  y8]  terms

[ 9, 1,37,43,  85],[ 28,  9, 25, 18, 57, 28, 50, 35],[ 27, 10, 27, 16, 55, 30, 53, 32], 0
[ 9, 5,83,85, 185],[ 53, 30, 42, 43,111, 74, 94, 91],[ 48, 35, 52, 33,101, 84,109, 76], 0
[13, 1,59,67, 135],[ 43, 16, 39, 28, 88, 47, 79, 56],[ 42, 17, 41, 26, 86, 49, 82, 53], 0
[13, 3,34,46,  80],[ 63,  5, 53, 39,125, 35, 97, 63],[ 57, 11, 65, 27,113, 47,115, 45], 0
[13, 5,69,75, 155],[ 50, 19, 39, 36,102, 53, 83, 72],[ 45, 24, 49, 26, 92, 63, 98, 57], 0
[13, 9,41,47,  85],[ 38,  3, 21, 26, 71, 14, 42, 43],[ 29, 12, 39,  8, 53, 32, 69, 16], 0
[17, 1,39,57,  95],[ 37,  2, 36, 21, 74, 21, 63, 32],[ 36,  3, 38, 19, 72, 23, 66, 29], 1
[17, 9,46,58, 100],[ 89,  3, 57, 59,169, 31,107, 93],[ 71, 21, 93, 23,133, 67,161, 39], 1
[17,13,31,41,  45],[ 39,  8, 16, 25, 61, 16, 20, 25],[ 26,  5, 42,  1, 35, 10, 59, 14], 0
[19, 1,33,57,  85],[ 36,  3, 37, 20, 72, 13, 60, 25],[ 35,  2, 39, 18, 70, 15, 63, 22], 0
[23,17,86,94, 180],[149, 23, 83,105,283, 77,175,185],[115, 57,151, 37,215,145,277, 83], 1
[25, 3,65,89, 155],[ 59,  6, 54, 35,118, 37, 98, 57],[ 56,  9, 60, 29,112, 43,107, 48], 0
[31, 3,47,89, 125],[ 56,  9, 57, 32,112, 13, 89, 36],[ 53,  6, 63, 26,106, 19, 98, 27], 0
[31,13, 3,69,  25],[ 39, 36, 37, 32, 72, 47, 24,  1],[ 26, 23, 63,  6, 46, 21, 63, 38], 0
[37, 1,96,132,230],[171, 21,167, 97,343,117,301,159],[169, 23,171, 93,339,121,307,153], 1
[37,17, 9,81,   5],[ 50, 41, 42, 39, 75, 70, 14,  9],[ 33, 24, 76,  5, 41, 36, 65, 60], 1
[39,17, 2,86,  20],[ 97, 93, 91, 81,171,131, 47,  7],[ 63, 59,159, 13,103, 63,149,109], 0
[39,29,73,97, 115],[ 90, 17, 39, 58,145, 30, 53, 62],[ 61, 12, 97,  0, 87, 28,140, 25], 0
[41,13,87,129,205],[ 91,  4, 72, 57,177, 28,124, 81],[ 78,  9, 98, 31,151, 54,163, 42], 0
[43, 9,76,128,190],[171, 19,153,103,337, 43,249,131],[153,  1,189, 67,301, 79,303, 77], 1
[43,29,66,102,100],[181, 49, 87,117,287, 87, 99,101],[123,  9,203,  1,171, 29,273, 73], 0
[45,31,65,103, 85],[ 93, 28, 43, 60,141, 56, 41, 44],[ 62,  3,105,  2, 79,  6,134, 49], 0
[46,24,47,107, 85],[163, 69,105,109,271,101,105, 65],[115, 21,201, 13,175,  5,249, 79], 0
[47,21, 1,103,  5],[ 58, 57, 54, 49, 94, 89, 18, 13],[ 37, 36, 96,  7, 52, 47, 81, 76], 0
[49,23,93,141,205],[107, 14, 72, 69,199,  6,117, 88],[ 84,  9,118, 23,153, 52,186, 19], 0
[51, 7,83,149,215],[ 96, 13, 93, 56,191, 24,148, 67],[ 89,  6,107, 42,177, 38,169, 46], 0
[53,17,96,156,230],[219, 27,175,137,423, 37,285,175],[185,  7,243, 69,355,105,387, 73], 0
[53,23,59,131,135],[ 94, 35, 69, 62,170, 35, 86, 49],[ 71, 12,115, 16,124, 11,155, 20], 0
[57,41,91,133,125],[123, 32, 54, 79,189, 64, 58, 67],[ 82,  9,136,  3,107, 18,181, 56], 0
[58,24,61,143,145],[201, 79,153,133,367, 77,189,101],[153, 31,249, 37,271, 19,333, 43], 1
[59,27,53,139,115],[ 99, 46, 72, 67,173, 58, 76, 39],[ 72, 19,126, 13,119,  4,157, 42], 1
[59,29,87,153,185],[117, 30, 77, 76,210, 25,108, 77],[ 88,  1,135, 18,152, 33,195, 10], 0
[59,37,67,131, 75],[111, 44, 58, 73,163, 88, 41, 34],[ 74,  7,132,  1, 89, 14,152, 77], 0
[59,39,77,133, 95],[117, 40, 57, 76,175, 80, 48, 47],[ 78,  1,135,  2, 97,  2,165, 70], 0
[61,27, 2,134, 20],[151,147,141,127,257,217, 61, 21],[ 97, 93,249, 19,149,109,223,183], 0
[61,37,98,154,180],[257, 61,141,167,437, 77,191,169],[183, 13,289, 19,289, 71,413, 53], 0
[61,43,93,141,125],[129, 36, 58, 83,197, 72, 59, 66],[ 86,  7,144,  3,111, 14,188, 63], 0
[67, 1,81,183,235],[108, 27,124, 59,219, 16,183, 52],[107, 26,126, 57,217, 18,186, 49], 0
[67,31,21,147, 25],[ 93, 72, 76, 71,144,119, 33,  8],[ 62, 41,138,  9, 82, 57,126,101], 0
[67,37,81,159,145],[126, 45, 76, 83,210, 65, 84, 61],[ 89,  8,150,  9,136,  9,195, 50], 0
[67,37,94,166,180],[265, 77,159,173,455, 95,203,157],[191,  3,307, 25,307, 53,425, 65], 0
[67,41,81,153,115],[128, 47, 69, 84,199, 84, 63, 52],[ 87,  6,151,  2,117,  2,186, 71], 0
[68,30,91,175,205],[257, 75,183,167,469, 59,251,159],[197, 15,303, 47,349, 61,431, 21], 0
[71,27, 7,161, 85],[ 88, 81, 89, 72,176, 91, 73, 12],[ 61, 54,143, 18,122, 37,154, 69], 0
[71,31,13,157, 45],[ 93, 80, 83, 74,160,115, 47,  2],[ 62, 49,145, 12, 98, 53,140, 95], 0
[71,31,22,158, 60],[195,151,167,149,335,215,109, 11],[133, 89,291, 25,211, 91,295,175], 0
[71,37,48,156, 50],[227,131,153,159,337,237, 81, 19],[153, 57,301, 11,189, 89,303,203], 1
[72,34,61,167,125],[239,117,171,163,409,159,167, 83],[171, 49,307, 27,273, 23,371,121], 1
[73, 7,89,199,255],[121, 32,129, 70,244, 11,190, 65],[114, 25,143, 56,230, 25,211, 44], 0
[73,37,39,159,  5],[111, 72, 79, 80,149,144, 20, 15],[ 74, 35,153,  6, 75, 70,131,126], 0
[77,33,61,181,145],[124, 63, 96, 85,221, 76,100, 45],[ 91, 30,162, 19,155, 10,199, 54], 0
[80,46,65,173, 15],[271,141,161,185,347,317, 37,  7],[179, 49,345,  1,163,133,313,283], 0
[81,29,67,197,185],[129, 62,110, 87,243, 58,130, 55],[100, 33,168, 29,185,  0,217, 32], 0
[81,29,83,203,215],[137, 54,113, 90,258, 43,145, 70],[108, 25,171, 32,200, 15,232, 17], 0
[81,43,52,176, 10],[257,153,171,181,339,319, 43, 23],[171, 67,343,  9,167,147,301,281], 1
[83,29,81,207,215],[138, 57,116, 91,261, 46,147, 68],[109, 28,174, 33,203, 12,234, 19], 0
[89,41,42,198, 80],[261,177,205,191,429,269,135, 25],[179, 95,369, 27,265,105,381,221], 0
[89,51,73,193, 35],[151, 78, 90,103,202,167, 30,  5],[100, 27,192,  1,100, 65,183,148], 0
[93,37, 1,209, 95],[112,111,114, 95,224,129, 85, 10],[ 75, 74,188, 21,150, 55,196,101], 0
[93,37,29,211,115],[126, 97,115, 96,234,119, 95, 20],[ 89, 60,189, 22,160, 45,206, 91], 0
[94,36,67,223,185],[291,157,245,201,539,169,265,105],[219, 85,389, 57,395, 25,481,111], 0
[94,48,67,209, 95],[303,169,207,211,473,283,139, 51],[207, 73,399, 19,281, 91,427,237], 1
[95,43,25,209, 45],[129,104,109,100,208,163, 53,  8],[ 86, 61,195, 14,122, 77,182,137], 0
[97,39,14,218,100],[247,219,237,199,469,269,177, 23],[169,141,393, 43,313,113,411,211], 0
[97,51,64,212, 50],[309,181,207,217,443,343, 91,  9],[207, 79,411, 13,239,139,397,297], 1