1.Introduction


Shinha's theorem,
      a1n + a2n + a3n = b1n + b2n + b3n, (n=2,4)  and
      a1 + a2 - a3 =2(b1 + b2 - b3), a1 + a2 <> a3, b1 + b2 <> b3
      then

      (2a1-3h)n + (2a1-h)n + (2a2-3h)n + (2a2-h)n + (2b3+h)n  = (2a3+h)n + (2a3+3h)n + (2b1-h)n + (2b2-h)n + (3h)n 
      (n=1,3,5,7)
      where 2h=b1 + b2 - b3.

By above theorem,Sinha(1966) gave a parametric solution for 7.5.5,

 (-7m2+62mn-30n2)7+(7m2+38mn-50n2)7+(5m2-8mn-22n2)7+(19m2-32mn-42n2)7+(-19m2+36mn-62n2)7
=(-9m2+66mn-42n2)7+(5m2+42mn-62n2)7+(-21m2+38mn-22n2)7+(9m2-14mn-50n2)7+(21m236mn-30n2)7.([1].Wolfram) 

Piezas showed there are four solutions of binary quadratic forms.([2].Piezas)

I add four parametric solutions besides the Sinha's one.

[1].Wolfram: Multigrade Eauation
[2].Tito Piezas:http://sites.google.com/site/tpiezas/026
 

2. Solutions

         I show only four solutions because there are many solutions.

         1. (x+3y+1)n + (x+y-1)n + (-6x-4y)n = (8x+3y)n + (6x-y)n + (10x-2y)n,(n=2,4)

            Denote sn=a1n+a2n+a3n-(b1n+b2n+b3n)

            s2 =-2(-6y2-2y-1-30xy+81x2)

            s4 =-2(-6y2-2y-1-30xy+81x2)(20y2+2y+1-6xy+87x2)

            So,if -6y2-2y-1-30xy+81x2 = 0 then this is a solution of 7.5.5 by Shinha's theorem.
   

           (-2x+1)n + (2y+1)n + (-2x-2y-1)n -1 + (11x-y)n 
           =
           (-5x-3y)n + (-3x-y)n + (7x+2y)n + (5x-2y)n + (3x+3y)n
           (n=1,3,5,7)   where -6y2-2y-1-30xy+81x2 = 0

           Parametric solution is 

           (42k2+538k-1299)n + (-202k2+1362k-1701)n + (166k2-1214k+1455)n + (-78k2-390k+1053)n + (338k2-1300k+1677)n 
           = 
           (330k2-1088k+357)n + (86k2-264k-45)n + (-154k2+454k+213)n + (370k2-1342k+1263)n + (-366k2+1236k-603)n
           (n=1,3,5,7) 

         2. (2x+y+1)n + (x+3y-1)n + (-9x-4y)n = (12x+3y)n + (9x-y)n + (15x-2y)n,(n=2,4)
            
            s2 = -2(-6y2+2y-1-44xy-x+182x2)

            s4 = -2(-6y2+2y-1-44xy-x+182x2)(20y2-2y+1-10xy+x+196x2)

            (-5x-4y+2)n + (x+2)n + (-7x-2)n + (-x+4y-2)n + (33x-2y)n
            = 
            (-15x-6y)n + (-9x-2y)n + (21x+4y)n + (15x-4y)n + (9x+6y)n
            (n=1,3,5,7)   where -6y2+2y-1-44xy-x+182x2 = 0

            Parametric solution is  

            (-1522-1042k+359k2)n + (-3918+942k+141k2)n + (3402-786k-195k2)n + (1006+1198k-413k2)n + (4294-1928k+433k2)n
            =
            (3078-2820k+273k2)n + (682-836k+55k2)n + (-1106+1594k-83k2)n + (4202-2530k+407k2)n + (-3594+2976k-327k2)n



         3. (4x+2y+1)^n + (x+2y-1)^n + (-7x-4y)^n = (7x+3y)^n + (2x-y)^n + (3x-2y)^n,(n=2,4)

            s2 = 4x^2+50xy+6x+10y^2+2

            s4 = 2(2x^2+25xy+3x+5y^2+1)(40x^2+3x+1+35xy+19y^2)

            (-x-2y+2)^n + (5x+2y+2)^n + (-7x-2y-2)^n + (-x+2y-2)^n + (9x-2y)^n 
            =
            (-11x-6y)^n + (-5x-2y)^n + (11x+4y)^n + (x-4y)^n + (9x+6y)^n
            (n=1,3,5,7)   where 2x^2+25xy+3x+5y^2+1 = 0

            Parametric solution is

            (5-35k^2+48k)^n + (-1+35k^2+52k)^n + (3-25k^2-52k)^n + (-3+45k^2-48k)^n + (-9-95k^2-2k)^n 
            =
            (11-95k^2-6k)^n + (5-25k^2-2k)^n + (-11+45k^2+4k)^n + (-1-105k^2-4k)^n + (-9+105k^2+6k)^n

         4. (7x+y+1)^n + (3x+3y-1)^n + (-14x-4y)^n = (14x+3y)^n + (4x-y)^n + (6x-2y)^n,(n=2,4)

            s2 = 2(3x^2+4x+1+46xy-2y+6y^2)
            s4 = 2(3x^2+4x+1+46xy-2y+6y^2)(155x^2+4x+1+66xy-2y+20y^2)
            
            (-2x-2y+1)^n + (4x+1)^n + (-6x-1)^n + (2y-1)^n + (9x-y)^n
            =
            (-11x-3y)^n + (-5x-y)^n + (11x+2y)^n + (x-2y)^n + (9x+3y)^n

            Parametric solution is

            (5+38k-78k^2)^n + (-1+54k-18k^2)^n + (3-58k+30k^2)^n + (-42k+90k^2-3)^n + (-9+16k-102k^2)^n
            =
            (11-28k-78k^2)^n + (5-12k-18k^2)^n + (-11+26k+30k^2)^n + (-1-2k-102k^2)^n + (-9+24k+90k^2)^n


3. Example

      Case 1.   n=1,3,5,7

      k
      
      1  -719n   -541n   +407n   +585n    +715n =  -401n   -223n   +513n   +291n     +267n
      2   -55n   +215n   -309n    -39n    +429n =  -499n   -229n   +505n    +59n     +405n
      3    11n     +9n    -11n    -13n     +13n =     1n     -1n     +3n     +9n       -3n
      4   305n   +103n   -149n   -351n    +377n =   257n    +55n    -87n   +363n     -303n
      5  2441n    +59n   -465n  -2847n   +3627n =  3167n   +785n  -1367n  +3803n    -3573n
      6  1147n   -267n    +49n  -1365n   +2015n =  1903n   +489n   -869n  +2177n    -2121n
      7  4525n  -2065n  +1091n  -5499n   +9139n =  8911n  +2321n  -4155n  +9999n    -9885n
      8  5693n  -3733n  +2367n  -7059n  +12909n = 12773n  +3347n  -6011n  +14207n  -14139n
      9   463n   -387n   +265n   -585n   +1157n =  1153n   +303n   -545n  +1277n    -1275n
     10     7n     -7n     +5n     -9n     +19n =    19n     +5n     -9n    +21n      -21n




      Case 2.

      k

      1  -245n   -315n   +269n   +199n   +311n =    59n    -11n    +45n   +231n   -105n 
      2   -31n    -21n    +15n    +25n    +31n =   -21n    -11n    +25n    +11n    +15n
      3 -1417n   +177n   -711n   +883n  +2407n = -2925n  -1331n  +2929n   +275n  +2391n
      4     1n    +39n    -53n    -15n    +65n =   -71n    -33n    +73n    +11n    +57n
      5  2243n  +4317n  -5403n  -3329n  +5479n = -4197n  -2123n  +4789n  +1727n  +3111n
      6  2575n  +3405n  -4167n  -3337n  +4157n = -2007n  -1177n  +2735n  +1837n  +1245n
      7   195n   +213n   -259n   -241n   +267n =   -73n    -55n   +133n   +143n    +27n
      8  6559n  +6321n  -7683n  -7921n  +8291n = -1005n  -1243n  +3167n  +5005n   -357n
      9  2597n  +2283n  -2781n  -3095n  +3145n =   -27n   -341n   +931n  +2057n   -471n
     10    11n     +9n    -11n    -13n    +13n =     1n     -1n     +3n     +9n     -3n


      Case 3.

      k

      1     9^n    +43^n    -37^n     -3^n    -53^n =   -45^n    -11^n    +19^n     -55^n      +51^n
      2   -13^n    +81^n    -67^n    +27^n   -131^n =  -127^n    -33^n    +59^n    -143^n     +141^n
      3   -83^n   +235^n   -189^n   +129^n   -435^n =  -431^n   -113^n   +203^n    -479^n     +477^n
      4  -363^n   +767^n   -605^n   +525^n  -1537^n = -1533^n   -403^n   +725^n   -1697^n    +1695^n
      5    -5^n     +9^n     -7^n     +7^n    -19^n =   -19^n     -5^n     +9^n     -21^n      +21^n
      6  -967^n  +1571^n  -1209^n  +1329^n  -3441^n = -3445^n   -907^n  +1633^n   -3805^n    +3807^n
      7  -687^n  +1039^n   -793^n   +933^n  -2339^n = -2343^n   -617^n  +1111^n   -2587^n    +2589^n
      8  -617^n   +885^n   -671^n   +831^n  -2035^n = -2039^n   -537^n   +967^n   -2251^n    +2253^n
      9 -1199^n  +1651^n  -1245^n  +1605^n  -3861^n = -3869^n  -1019^n  +1835^n   -4271^n    +4275^n
     10 -3015^n  +4019^n  -3017^n  +4017^n  -9529^n = -9549^n  -2515^n  +4529^n  -10541^n   +10551^n


      Case 4.

      k


      1    -7^n     +7^n     -5^n     +9^n     -19^n =   -19^n     -5^n     +9^n     -21^n    +21^n
      2   -33^n     +5^n     +1^n    +39^n     -55^n =   -51^n    -13^n    +23^n     -59^n    +57^n
      3  -583^n     -1^n    +99^n   +681^n    -879^n =  -775^n   -193^n   +337^n    -925^n   +873^n
      4 -1091^n    -73^n   +251^n  +1269^n   -1577^n = -1349^n   -331^n   +573^n   -1641^n  +1527^n
      5 -1755^n   -181^n   +463^n  +2037^n   -2479^n = -2079^n   -505^n   +869^n   -2561^n  +2361^n
      6  -515^n    -65^n   +147^n   +597^n    -717^n =  -593^n   -143^n   +245^n    -737^n   +675^n
      7 -3551^n   -505^n  +1067^n  +4113^n   -4895^n = -4007^n   -961^n  +1641^n   -5013^n  +4569^n
      8 -4683^n   -721^n  +1459^n  +5421^n   -6409^n = -5205^n  -1243^n  +2117^n   -6545^n  +5943^n
      9  -853^n   -139^n   +273^n   +987^n   -1161^n =  -937^n   -223^n   +379^n   -1183^n  +1071^n
     10 -7415^n  -1261^n  +2423^n  +8577^n  -10049^n = -8069^n  -1915^n  +3249^n  -10221^n  +9231^n