1.Introduction




I found many parameter solutions for A15+ A25+ A35+ A45+ A55+ A65+ A75 = B15.

Actually, the positive value gives the solution of the 5.4.4 and 5.3.5 identity.

2. Theorem
          
         There are  many parameter solutions for 5.7.1 equation. 
         
          A15+ A25+ A35+ A45+ A55+ A65+ A75 = B15

          a1,a2,c,m,n: integer

          A1= c+a1
          A2= c-a1
          A3= c+a2
          A4= c-a2
          A5= c+(a1+a2)
          A6= c-(a1+a2)
          A7= n
          B1= m
         


         Condition
  
         20c(a12+a22+a1a2)(a12+a1a2+2c2+a22)+6c5 = m5-n5


Proof.

(c+a1)5+(c-a1)5+(c+a2)5+(c-a2)5+(c+a1+a2)5+(c-a1-a2)5
=20c(a12+a22+a1a2)(a12+a1a2+2c2+a22)+6c5............................(1)

If there are m,n which satisfy 20c(a12+a22+a1a2)(a12+a1a2+2c2+a22)+6c5 = m5-n5,
we can obtain the parameter solution.


Let C be a rational conic with equation f(a1,a2)=a12+a1a2+a22.
Take P(a10,a20) a rational point on C,then we can get many rational points infinitely.
We parametrise the rational point on C.
Let a1=a10+t,a2=a20+kt...........................................................(2)
then substitute (2) to a12+a1a2+a22=a102+a10a20+a202.
Solve for t,and obtain t=g(k)....................................................(3)
g(k) is rational function of t.
Substitute (3) to (2),then

a1=a10+g(k)=ha(k)
a2=a20+kg(k)=hb(k)...............................................................(4)


Substitute (4) to (1),then obtain a parameter solution.

There are many (c,a1,a2,m,n) which satisfy 20c(a12+a22+a1a2)(a12+a1a2+2c2+a22)+6c5 = m5-n5
like the next example.


Q.E.D.

3. Example

(c,a1,a2,m,n)=( 7,36,2,49,7)
(43k2+3k-31)5+(-29k2+11k+45)5+(-31k2-65k+9)5+(45k2+79k+5)5+(5k2-69k-29)5+(9k2+83k+43)5+(7k2+7k+7)5
=(49k2+49k+49)5

(c,a1,a2,m,n)=( 13,58,18,91,13)
(71k2-23k-63)5+(-45k2+49k+89)5+(-63k2-103k+31)5+(89k2+129k-5)5+(-5k2-139k-45)5+(31k2+165k+71)5+(13k2+13k+13)5
=(91k2+91k+91)5

(c,a1,a2,m,n)=( 13,62,12,91,13)
(75k2-11k-61)5+(-49k2+37k+87)5+(-61k2-111k+25)5+(87k2+137k+1)5+(k2-135k-49)5+(25k2+161k+75)5+(13k2+13k+13)5
=(91k2+91k+91)5

(c,a1,a2,m,n)=(31,114,74,217,31)
(145k2-117k-157)5+(-83k2+179k+219)5+(-157k2-197k+105)5+(219k2+259k-43)5+(-43k2-345k-83)5+(105k2+407k+145)5+(31k2+31k+31)5
=(217k2+217k+217)5

(c,a1,a2,m,n)=( 31,162,4,217,31)
(193k2+23k-135)5+(-131k2+39k+197)5+(-135k2-293k+35)5+(197k2+355k+27)5+(27k2-301k-131)5+(35k2+363k+193)5+(31k2+31k+31)5
=(217k2+217k+217)5

(c,a1,a2,m,n)=(19,62,54,133,19)
(81k2-89k-97)5+(-43k2+127k+135)5+(-97k2-105k+73)5+(135k2+143k-35)5+(-35k2-213k-43)5+(73k2+251k+81)5+(19k2+19k+19)5
=(133k2+133k+133)5

(c,a1,a2,m,n)=(19,94,12,133,19)
(113k2-5k-87)5+(-75k2+43k+125)5+(-87k2-169k+31)5+(125k2+207k+7)5+(7k2-193k-75)5+(31k2+231k+113)5+(19k2+19k+19)5
=(133k2+133k+133)5

(c,a1,a2,m,n)=(37,118,108,259,37)
(155k2-179k-189)5+(-81k2+253k+263)5+(-189k2-199k+145)5+(263k2+273k-71)5+(-71k2-415k-81)5+(145k2+489k+155)5+(37k2+37k+37)5
=(259k2+259k+259)5

(c,a1,a2,m,n)=(37,174,38,259,37)
(211k2-39k-175)5+(-137k2+113k+249)5+(-175k2-311k+75)5+(249k2+385k-1)5+(-k2-387k-137)5+(75k2+461k+211)5+(37k2+37k+37)5
=(259k2+259k+259)5

(c,a1,a2,m,n)=(43,148,114,301,43)
(191k2-185k-219)5+(-105k2+271k+305)5+(-219k2-253k+157)5+(305k2+339k-71)5+(-71k2-481k-105)5+(157k2+567k+191)5+(43k2+43k+43)5
=(301k2+301k+301)5

(c,a1,a2,m,n)=( 49,174,124,343,49)
(223k2-199k-249)5+(-125k2+297k+347)5+(-249k2-299k+173)5+(347k2+397k-75)5+(-75k2-547k-125)5+(173k2+645k+223)5+(49k2+49k+49)5
=(343k2+343k+343)5




Numerical example

1<=k<=20

(43k2+3k-31)5+(-29k2+11k+45)5+(-31k2-65k+9)5+(45k2+79k+5)5+(5k2-69k-29)5+(9k2+83k+43)5+(7k2+7k+7)5
=(49k2+49k+49)5

       k

     ( 1)      55    +95   -295   +435   -315   +455    +75=    495
     ( 2)      35    -15    -55    +75    -35    +55    +15=     75
     ( 3)    3655  -1835  -4655  +6475  -1915  +3735   +915=   6375
     ( 4)    2235  -1255  -2495  +3475   -755  +1735   +495=   3435
     ( 5)   10595  -6255 -10915 +15255  -2495  +6835  +2175=  15195
     ( 6)   15355  -9335 -14975 +20995  -2635  +8655  +3015=  21075
     ( 7)    6995  -4335  -6555  +9215   -895  +3555  +1335=   9315
     ( 8)   27455 -17235 -24955 +35175  -2615 +12835  +5115=  35775
     ( 9)     715   -455   -635   +895    -55   +315   +135=    915
     (10)   14335  -9155 -12475 +17655   -735  +5915  +2595=  18135
     (11)   52055 -33435 -44575 +63195  -1835 +20455  +9315=  65175
     (12)   61975 -39995 -52355 +74335  -1375 +23355 +10995=  76935
     (13)   24255 -15715 -20255 +28795   -275  +8815  +4275=  29895
     (14)   84395 -54855 -69775 +99315   -155 +29695 +14775= 103395
     (15)   96895 -63155 -79415+113155   +615 +33135 +16875= 118095
     (16)     755   -495   -615   +875    +15   +255   +135=    915
     (17)  124475 -81495-100555+143535  +2435 +40555 +21495= 150435
     (18)  139555 -91535-112055+160075  +3495 +44535 +24015= 168075
     (19)   51835 -34055 -41395 +59175  +1555 +16235  +8895=  62235
     (20)  172295-113355-136915+195855  +5915 +53035 +29475= 206295