1.Introduction

Sinha(1966) gave a parameter 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) 

This identity is obtained by solving below equation.

(a+2b+c)7 + (a+2b-c)7 + (a-b)7  + (2a+9b)7 + (3a+b)7=  (a+c)7 + (a-c)7 +(a+13b)7 + (2a-3b)7 + (3a+3b)7([2].Piezas)

I show that there is a parameter solution besides the Sinha's one.

[1].Wolfram: Multigrade Eauation
[2].Titus Piezas: Binary Quadratic Forms as Equal Sums of Like Powers

2. Theorem


             There is a parameter solution of A17+ A27+ A37+ A47+ A57 = B17+ B27+ B37+ B47+ B57.

             A17+ A27+ A37+ A47+ A57 = B17+ B27+ B37+ B47+ B57.................(1)


             A1=45-184k+177k2
             A2=25-152k+201k2
             A3=35-92k+39k2
             A4=105-426k+423k2
             A5=95-382k+393k2
             B1=35-192k+243k2
             B2=45-132k+81k2
             B3=25-102k+99k2
             B4=105-424k+429k2
             B5=95-386k+381k2



Proof.            


             A1=a+2b+1
             A2=a+2b-1
             A3=a-11b
             A4=2a+7b
             A5=3a+3b 
             B1=a+1
             B2=a-1
             B3=a+3b
             B4=2a-5b
             B5=3a+5b


      A17+ A27+ A37+ A47+ A57 -( B17+ B27+ B37+ B47+ B57)

     =-28b(-5a2-1-30ab+81b2)(8229b4-2910ab3+101b2+425a2b2-240a3b+1+10a2-35a4)


    
    5a2+30ab-81b2=-1.............................(2)

    We can find infinitely many rational solutions of (2),
    then we obtain infinitely rational solutions of (1).
   

    (a,b)=(2,1) is a solution of (2).

    We obtain parameter solution of (2) by using (a,b)=(2,1).


             A1=45-184k+177k2
             A2=25-152k+201k2
             A3=35-92k+39k2
             A4=105-426k+423k2
             A5=95-382k+393k2
             B1=35-192k+243k2
             B2=45-132k+81k2
             B3=25-102k+99k2
             B4=105-424k+429k2
             B5=95-386k+381k2






   
3. Example

         k
      
         1    197+   377+   -97+   517+   537=   437+   -37+   117+   557+   457

         2    557+   757+    17+  1357+  1297=   897+   157+   317+  1397+  1217

         3   5437+  6897+   557+ 13177+ 12437=  8237+  1897+  3057+ 13477+ 11837

         4  21417+ 26337+  2917+ 51697+ 48557= 31557+  8137+ 12017+ 52737+ 46477

         5   3557+  4297+   557+  8557+  8017=  5157+  1417+  1997+  8717+  7697

         6  53137+ 63497+  8877+127777+119517= 76317+ 21697+ 29777+130057+114957

         7  37157+ 44057+  6517+ 89257+ 83397= 52997+ 15457+ 20817+ 90797+ 80317

         8  99017+116737+ 17957+237697+221917=140517+ 41737+ 55457+241697+213917

         9   9097+ 10677+  1697+ 21817+ 20357= 12857+  3877+  5097+ 22177+ 19637

        10  31817+ 37217+  6037+ 76297+ 71157= 44837+ 13657+ 17817+ 77537+ 68677






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