1.Introduction


I show the collection of parametric solutions for x1^k + x2^k + x3^k = y1^k + y2^k + y3^k where k = 5.


I don't know whether there is a parametric solution that satisfies only k=5 excluding the case of Sastry and Moessner.


2.Parametric solutions
        
     
      
　1. k=5
     
    1.1 Degree 5

      (Sastry)

      x1 = a^5 + 25b^5
      x2 = a^5 - 25b^5
      x3 = 10a^3b^2

      y1 = a^5 + 75b^5
      y2 = a^5 - 75b^5
      y3 = -50ab^4

    1.2 Degree 36

      (Moessner)

      x1 = a^36 + 8a^26 + 12a^16 + 20a^11 - a^6
      x2 = a^30 + 20a^25 - 12a^20 - 8a^10 - 1
      x3 = a^33 - 12a^23 - 28a^13 - a^3

      y1 = a^36 + 8a^26 + 12a^16 - 20a^11 - a^6
      y2 = a^30 - 20a^25 - 12a^20 - 8a^10 - 1
      y3 = a^33 + 28a^23 + 12a^13 - a^3

      
　2. k=1,5
     

   2.1 Degree 7

       (Bremner[1])

      x1 = 75a^7 - 230a^6 - 113a^5 + 510a^4 - 407a^3 + 62a^2 + 125a - 150
      x2 = -75a^7 - 70a^6 + 153a^5 + 54a^4 - 217a^3 + 606a^2 - 245a + 50
      x3 = -175a^7 + 170a^6 - 391a^5 - 30a^4 + 451a^3 - 602a^2 + 115a - 50

      y1 = -25a^7 - 653a^5 + 564a^4 - 195a^3 - 208a^2 + 105a - 100
      y2 = -175a^7 + 160a^6 - 387a^5 + 108a^4 - 5a^3 + 336a^2 - 265a + 100
      y3 = 25a^7 - 290a^6 + 689a^5 - 138a^4 + 27a^3 - 62a^2 + 155a - 150


      (Bremner,Choudhry[2],Tomita[4])

      x1 = 5a^7 + 6a^6 + 51a^5 + 58a^4 + 108a^3 + 96a^2 + 36a + 40
      x2 = 5a^7 - 10a^6 + 39a^5 - 90a^4 + 52a^3 - 160a^2 + 4a - 40
      x3 = -5a^7 + 4a^6 - 39a^5 + 32a^4 - 52a^3 + 64a^2 - 4a

      y1 = -5a^7 - 4a^6 - 39a^5 - 32a^4 - 52a^3 - 64a^2 - 4a
      y2 = 5a^7 - 6a^6 + 51a^5 - 58a^4 + 108a^3 - 96a^2 + 36a - 40
      y3 = 5a^7 + 10a^6 + 39a^5 + 90a^4 + 52a^3 + 160a^2 + 4a + 40

      (Choudhry)

      x1 = -4a^7 - 17a^6 + 45a^5 - 10a^4 + 112a^3 - 124a^2 - 28a - 24
      x2 = 8a^7 + 9a^6 + 57a^5 - 32a^4 - 8a^3 - 68a^2 + 68a + 16
      x3 = 2a^7 + 29a^6 + 45a^5 + 66a^4 + 24a^3 + 156a^2 + 4a + 24

      y1 = 2a^7 - 3a^6 + 45a^5 - 66a^4 + 24a^3 - 164a^2 + 4a + 8
      y2 = -4a^7 + 17a^6 + 33a^5 + 32a^4 + 56a^3 + 60a^2 - 60a + 16
      y3 = 8a^7 + 7a^6 + 69a^5 + 58a^4 + 48a^3 + 68a^2 + 100a - 8


      (Tomita)

      x1 = 100a^7 + 145ba^6 + 243b^2a^5 + 250b^3a^4 + 282b^4a^3 + 165b^5a^2 + 95b^6a
      x2 = 50a^7 + 15ba^6 + 7b^2a^5 - 90b^3a^4 - 32b^4a^3 - 5b^5a^2 + 55b^6a
      x3 = -100a^7 - 135ba^6 - 211b^2a^5 - 176b^3a^4 - 234b^4a^3 - 199b^5a^2 - 175b^6a - 50b^7

      y1 = 50a^7 + 175ba^6 + 199b^2a^5 + 234b^3a^4 + 176b^4a^3 + 211b^5a^2 + 135b^6a + 100b^7
      y2 = -95ba^6 - 165b^2a^5 - 282b^3a^4 - 250b^4a^3 - 243b^5a^2 - 145b^6a - 100b^7
      y3 = -55ba^6 + 5b^2a^5 + 32b^3a^4 + 90b^4a^3 - 7b^5a^2 - 15b^6a - 50b^7

      This identity has a interesting property.
      Let f1(a,b)=x1,f2(a,b)=x2, and f3(a,b)=x3,then y1=-f3(b,a),y2=-f1(b,a),y3=-f2(b,a).


   2.2 Degree 8

      (Bremner)

      x1 = 389a^8 + 1111a^7 + 1599a^6 + 1435a^5 + 897a^4 + 417a^3 + 137a^2 + 29a + 2
      x2 = 123a^8 + 41a^7 + 129a^6 + 421a^5 + 543a^4 + 319a^3 + 87a^2 + 3a - 2
      x3 = 442a^8 + 1647a^7 + 2603a^6 + 2363a^5 + 1275a^4 + 409a^3 + 77a^2 + 13a + 3

      y1 = 272a^8 + 729a^7 + 1425a^6 + 1685a^5 + 1221a^4 + 527a^3 + 135a^2 + 19a + 3
      y2 = 474a^8 + 1655a^7 + 2647a^6 + 2363a^5 + 1263a^4 + 385a^3 + 49a^2 - 3a - 1
      y3 = 208a^8 + 415a^7 + 259a^6 + 171a^5 + 231a^4 + 233a^3 + 117a^2 + 29a + 1


      (Tomita)

      x1 = 2a^8 + a^7 + 20a^6 + 29a^5 + 74a^4 + 84a^3 + 64a^2 + 36a + 40
      x2 = 7a^7 - 8a^6 + 45a^5 - 74a^4 + 44a^3 - 128a^2 + 4a - 40
      x3 = -2a^8 - 3a^7 - 12a^6 - 41a^5 - 84a^3 + 64a^2 + 28a

      y1 = 7a^7 + 8a^6 + 45a^5 + 74a^4 + 44a^3 + 128a^2 + 4a + 40
      y2 = -2a^8 + a^7 - 20a^6 + 29a^5 - 74a^4 + 84a^3 - 64a^2 + 36a - 40
      y3 = 2a^8 - 3a^7 + 12a^6 - 41a^5 - 84a^3 - 64a^2 + 28a


      (Tomita)

      x1 = -a^8 - 5a^7 + 5a^6 - 25a^5 + 70a^4 - 52a^3 + 140a^2 + 44a + 24
      x2 = -3a^8 - a^7 - 17a^6 - 33a^5 - 78a^4 - 52a^3 - 172a^2 - 20a - 24
      x3 = a^8 + 9a^7 + 3a^6 + 49a^5 + 62a^4 + 84a^3 + 116a^2 + 84a - 8

      y1 = -3a^8 - a^7 - 15a^6 - 21a^5 - 4a^4 - 28a^3 - 20a^2 + 76a + 16
      y2 = a^8 - a^7 + 7a^6 - 33a^5 + 54a^4 - 84a^3 + 84a^2 - 20a - 8
      y3 = -a^8 + 5a^7 - a^6 + 45a^5 + 4a^4 + 92a^3 + 20a^2 + 52a - 16






    2.3 Degree 9

      (Choudhry)

      x1 = a - a^3 - 2a^5 + a^9
      x2 = 1 + a^2 - 2a^6 + 2a^7 + a^8
      x3 = 2a^3 + 2a^4 - 2a^7

      y1 = a + 3a^3 - 2a^5 + a^9
      y2 = 1 + a^2 - 2a^6 - 2a^7 + a^8
      y3 = -2a^3 + 2a^4 + 2a^7

      (Lander[3])

      x1 = -2a^8 + 10a^7 + 20a^6 + 20a^5 + 34a^4 - 10a^3 - 270a^2 - 20a - 682
      x2 = 2a^8 + 10a^7 - 20a^6 + 20a^5 - 34a^4 - 10a^3 + 270a^2 - 20a + 682
      x3 = a^9 - 22a^5 - 125a^3 - 79a

      y1 = a^8 + 10a^7 - 10a^6 + 20a^5 - 92a^4 - 160a^3 - 15a^2 - 320a + 341
      y2 = a^9 - 22a^5 + 175a^3 + 521a
      y3 = -a^8 + 10a^7 + 10a^6 + 20a^5 + 92a^4 - 160a^3 + 15a^2 - 320a - 341

      Lander obtained three-parameter solution of degree 9,this example is a special case with (p,q)=(-2,1).

      (Tomita)

      x1 = a^9 + 6a^8 - a^7 - 28a^6 - 53a^5 - 50a^4 - 27a^3 - 8a^2 - a
      x2 = 2a^9 + 2a^8 - 8a^7 - 20a^6 - 20a^5 - 10a^4 - 2a^3
      x3 = -2a^9 - 18a^8 - 48a^7 - 60a^6 - 40a^5 - 14a^4 - 2a^3

      y1 = a^9 - 7a^8 - 49a^7 - 117a^6 - 161a^5 - 147a^4 - 91a^3 - 37a^2 - 9a - 1
      y2 = -3a^9 - 18a^8 - 61a^7 - 108a^6 - 113a^5 - 74a^4 - 31a^3 - 8a^2 - a
      y3 = 3a^9 + 15a^8 + 53a^7 + 117a^6 + 161a^5 + 147a^4 + 91a^3 + 37a^2 + 9a + 1



    2.4 Degree 10

      (Choudhry)

      x1 = 2a^10 + a^9 + 28a^8 + 33a^7 + 154a^6 + 200a^5 + 360a^4 + 372a^3 + 296a^2 + 144a + 160
      x2 = 7a^9 - 8a^8 + 73a^7 - 106a^6 + 224a^5 - 424a^4 + 180a^3 - 552a^2 + 16a - 160
      x3 = -2a^10 - 3a^9 - 20a^8 - 53a^7 - 48a^6 - 248a^5 + 64a^4 - 308a^3 + 256a^2 + 112a

      y1 = 7a^9 + 8a^8 + 73a^7 + 106a^6 + 224a^5 + 424a^4 + 180a^3 + 552a^2 + 16a + 160
      y2 = 2a^10 - 3a^9 + 20a^8 - 53a^7 + 48a^6 - 248a^5 - 64a^4 - 308a^3 - 256a^2 + 112a
      y3 = -2a^10 + a^9 - 28a^8 + 33a^7 - 154a^6 + 200a^5 - 360a^4 + 372a^3 - 296a^2 + 144a - 160

    2.5 Degree 12

      (Tomita)

      x1 = 2a^12 - a^11 + 16a^10 - 11a^9 + 52a^8 + 2a^7 + 40a^6 - 10a^5 + 42a^4 + 15a^3 + 8a^2 + 5a
      x2 = 5a^11 - 8a^10 + 15a^9 - 42a^8 - 10a^7 - 40a^6 + 2a^5 - 52a^4 - 11a^3 - 16a^2 - a - 2
      x3 = -2a^12 + 3a^11 - 8a^10 + 27a^9 - 10a^8 + 50a^7 + 50a^5 + 10a^4 + 27a^3 + 8a^2 + 3a + 2

      y1 = 2a^12 + 3a^11 + 8a^10 + 27a^9 + 10a^8 + 50a^7 + 50a^5 - 10a^4 + 27a^3 - 8a^2 + 3a - 2
      y2 = 5a^11 + 8a^10 + 15a^9 + 42a^8 - 10a^7 + 40a^6 + 2a^5 + 52a^4 - 11a^3 + 16a^2 - a + 2
      y3 = -2a^12 - a^11 - 16a^10 - 11a^9 - 52a^8 + 2a^7 - 40a^6 - 10a^5 - 42a^4 + 15a^3 - 8a^2 + 5a


    2.6 Degree 13

      (Tomita)

      x1 = 2a^12 + 2a^11 - a^10 + 3a^9 + 12a^8 + 4a^7 + a^6 + 13a^5 + 10a^4 + 3a^2 + a
      x2 = -a^12 - a^9 + 2a^8 + 6a^7 + 2a^6 + a^5 + 8a^4 + 6a^3 + a + 1
      x3 = -a^13 - a^12 + a^11 - 2a^10 - 7a^9 - 4a^8 - 5a^7 - 8a^6 - 9a^5 - 6a^4 - 5a^3 - 2a^2 - 1

      y1 = a^12 + 7a^9 + 6a^8 - 2a^7 + 6a^6 + 9a^5 - 2a^3 + a - 1
      y2 = -a^13 - a^12 + a^11 - 2a^10 - 7a^9 + 2a^8 + 3a^7 - 4a^6 - a^5 + 4a^4 + 3a^3 + 2a^2 + 1
      y3 = 2a^11 - a^10 - 5a^9 + 2a^8 + 4a^7 - 7a^6 - 3a^5 + 8a^4 - a^2 + a

    2.7 Degree 14

      (Tomita)

      x1 = 8a^14 - 16a^13 - 4a^12 + 32a^11 - 4a^10 - 32a^9 + 12a^8 + 40a^7 - 2a^6 - 8a^5 + 25a^4 + 16a^3 - 7a^2 - 8a - 2
      x2 = 4a^12 - 20a^10 + 24a^9 + 12a^8 - 48a^7 + 4a^6 + 28a^5 - 17a^4 - 12a^3 + 13a^2 + 10a + 2
      x3 = -8a^14 + 16a^13 + 4a^12 - 32a^11 + 4a^10 + 32a^9 - 12a^8 - 24a^7 + 10a^6 + 8a^5 - 13a^4 + 11a^2 + 4a

      y1 = 8a^13 - 12a^12 + 12a^10 + 4a^8 - 8a^7 + 4a^6 + 30a^5 + 15a^4 + 12a^3 + 21a^2 + 12a + 2
      y2 = -8a^13 + 12a^12 + 16a^11 - 36a^10 - 8a^9 + 36a^8 - 8a^7 - 36a^6 - 6a^5 - 3a^4 - 24a^3 - 23a^2 - 10a - 2
      y3 = 4a^12 - 16a^11 + 4a^10 + 32a^9 - 28a^8 - 16a^7 + 44a^6 + 4a^5 - 17a^4 + 16a^3 + 19a^2 + 4a





3.References

      1. A. Bremner, A geometric approach to equal sums of fifth powers,J. Number Theory, 13 (1981).

      2. Ajai Choudhry, On equal sums of fifth powers,Indian J. pure appl. Math., 28(1997)

      3. L.J.Lander,Geometric aspects of diophantine equations involving equal sums of powers,
         Amer. Math. Monthly 75 (1968)

      4. The parametric solution for equal sums of fifth powers