1.Introduction

I found many parameter solutions for A15+ A25+ A35+ A45+ A55+ A65+ A75 = 2.


2. Result
          
         There are many parameter solutions for the equation.
         
         A15+ A25+ A35+ A45+ A55+ A65+ A75 = 2.



(1+270x5)5+(1-270x5)5+(1+360x5)5+(1-360x5)5+(180x4)5-(1+450x5)5-(1-450x5)5=2

(1+11250x5)5+(1-11250x5)5+(1+27000x5)5+(1-27000x5)5+(4500x4)5-(1+29250x5)5-(1-29250x5)5=2

(1+135000x5)5+(1-135000x5)5+(1+72000x5)5+(1-72000x5)5+(18000x4)5-(1+153000x5)5-(1-153000x5)5=2

(1+123480x5)5+(1-123480x5)5+(1+423360x5)5+(1-423360x5)5+(35280x4)5-(1+441000x5)^5-(1-441000x5)5=2

(1+864000x5)5+(1-864000x5)5+(1+360000x5)5+(1-360000x5)5+(72000x4)5-(1+936000x5)^5-(1-936000x5)5=2

(1+729000x5)5+(1-729000x5)5+(1+3240000x5)5+(1-3240000x5)5+(162000x4)5-(1+3321000x5)5-(1-3321000x5)5=2

(1+2315250x5)5+(1-2315250x5)5+(1+2205000x5)5+(1-2205000x5)5+(220500x4)5-(1+3197250x5)5-(1-3197250x5)5=2

(1+2304000x5)5+(1-2304000x5)5+(1+4320000x5)5+(1-4320000x5)5+(288000x4)5-(1+4896000x5)5-(1-4896000x5)5=2

(1+3858750x5)5+(1-3858750x5)5+(1+1323000x5)5+(1-1323000x5)5+(220500x4)5-(1+4079250x5)5-(1-4079250x5)5=2

(1+2994750x5)5+(1-2994750x5)5+(1+16335000x5)5+(1-16335000x5)5+(544500x4)5-(1+16607250x5)5-(1-16607250x5)5=2



3. Method

(1+mx5)5+(1-mx5)5-(1+nx5)5-(1-nx5)5-(1+qx5)5-(1-qx5)5

=(10m4-10n4-10q4)x20+(20m2-20n2-20q2)x10-2................(1)

To do to make it to 5th powers the coefficient of x, m,n,q is decided. 

For example,first solution is (m,n,q)=(450,360,270),then we obtain

10m4-10n4-10q4=1805
20m2-20n2-20q2=0..........................................(2)

Substitute (m,n,q)=(450,360,270) and (2) to (1), and obtain 
(1+270x5)5+(1-270x5)5+(1+360x5)5+(1-360x5)5+(180x4)5-(1+450x5)5-(1-450x5)5=2


      I think that the parameter solution exists in the infinity.

4. Example

 (1+270x5)5+(1-270x5)5+(1+360x5)5+(1-360x5)5+(180x4)5-(1+450x5)5-(1-450x5)5=2

  1<=x<=20

  x       
                                               
  1        361^5        -359^5      +271^5      -269^5      +180^5       -451^5       +449^5=2
  2      11521^5      -11519^5     +8641^5     -8639^5     +2880^5     -14401^5     +14399^5=2
  3      87481^5      -87479^5    +65611^5    -65609^5    +14580^5    -109351^5    +109349^5=2
  4     368641^5     -368639^5   +276481^5   -276479^5    +46080^5    -460801^5    +460799^5=2
  5    1125001^5    -1124999^5   +843751^5   -843749^5   +112500^5   -1406251^5   +1406249^5=2
  6    2799361^5    -2799359^5  +2099521^5  -2099519^5   +233280^5   -3499201^5   +3499199^5=2
  7    6050521^5    -6050519^5  +4537891^5  -4537889^5   +432180^5   -7563151^5   +7563149^5=2
  8   11796481^5   -11796479^5  +8847361^5  -8847359^5   +737280^5  -14745601^5  +14745599^5=2
  9   21257641^5   -21257639^5 +15943231^5 -15943229^5  +1180980^5  -26572051^5  +26572049^5=2
 10   36000001^5   -35999999^5 +27000001^5 -26999999^5  +1800000^5  -45000001^5  +44999999^5=2
 11   57978361^5   -57978359^5 +43483771^5 -43483769^5  +2635380^5  -72472951^5  +72472949^5=2
 12   89579521^5   -89579519^5 +67184641^5 -67184639^5  +3732480^5 -111974401^5 +111974399^5=2
 13  133665481^5  -133665479^5+100249111^5-100249109^5  +5140980^5 -167081851^5 +167081849^5=2
 14  193616641^5  -193616639^5+145212481^5-145212479^5  +6914880^5 -242020801^5 +242020799^5=2
 15  273375001^5  -273374999^5+205031251^5-205031249^5  +9112500^5 -341718751^5 +341718749^5=2
 16  377487361^5  -377487359^5+283115521^5-283115519^5 +11796480^5 -471859201^5 +471859199^5=2
 17  511148521^5  -511148519^5+383361391^5-383361389^5 +15033780^5 -638935651^5 +638935649^5=2
 18  680244481^5  -680244479^5+510183361^5-510183359^5 +18895680^5 -850305601^5 +850305599^5=2
 19  891395641^5  -891395639^5+668546731^5-668546729^5 +23457780^5-1114244551^5+1114244549^5=2
 20 1152000001^5 -1151999999^5+864000001^5-863999999^5 +28800000^5-1440000001^5+1439999999^5=2




 (1+11250x5)5+(1-11250x5)5+(1+27000x5)5+(1-27000x5)5+(4500x4)5-(1+29250x5)5-(1-29250x5)5=2
         
  1<=x<=10
  x       

  1      11251^5      -11249^5     +27001^5     -26999^5     +4500^5     -29251^5     +29249^5=2
  2     360001^5     -359999^5    +864001^5    -863999^5    +72000^5    -936001^5    +935999^5=2
  3    2733751^5    -2733749^5   +6561001^5   -6560999^5   +364500^5   -7107751^5   +7107749^5=2
  4   11520001^5   -11519999^5  +27648001^5  -27647999^5  +1152000^5  -29952001^5  +29951999^5=2
  5   35156251^5   -35156249^5  +84375001^5  -84374999^5  +2812500^5  -91406251^5  +91406249^5=2
  6   87480001^5   -87479999^5 +209952001^5 -209951999^5  +5832000^5 -227448001^5 +227447999^5=2
  7  189078751^5  -189078749^5 +453789001^5 -453788999^5 +10804500^5 -491604751^5 +491604749^5=2
  8  368640001^5  -368639999^5 +884736001^5 -884735999^5 +18432000^5 -958464001^5 +958463999^5=2
  9  664301251^5  -664301249^5+1594323001^5-1594322999^5 +29524500^5-1727183251^5+1727183249^5=2
 10 1125000001^5 -1124999999^5+2700000001^5-2699999999^5 +45000000^5-2925000001^5+2924999999^5=2