1.Introduction

By Tito Piezas[1], it seems that x1^7+x2^7+x3^7+x4^7-x5^7-x6^7-x7^7-x8^7 = N has an infinitely many solutions,

Choudhry's Seventh Powers Theorem: “Any non-zero rational number N is the sum of eight rational 7th powers in an infinite number of non-trivial ways”  

We show that above theorem icnludes the parametric solution of x1^7 + x2^7 + x3^7 + x4^7 = y1^7 + y2^7 + y3^7 + y4^7 + y5^7.



2.Another view of Choudhry's Seventh Powers Theorem 
       
     
There is a parametric solution for x1^7 + x2^7 + x3^7 + x4^7 = y1^7 + y2^7 + y3^7 + y4^7 + y5^7.

     
Proof.

(x+a)^7+(x-a)^7+(mx+b)^7+(mx-b)^7-(x+c)^7-(x-c)^7-(mx+d)^7-(mx-d)^7= n^7.......(1)

Simplifying (1), we obtain

(-42d^2m^5-42c^2+42b^2m^5+42a^2)x^5
+(-70m^3d^4-70c^4+70m^3b^4+70a^4)x^3
+(-14c^6-14d^6m+14b^6m+14a^6)x= n^7............................................(2)


Decide coefficient of x^5 and x^3 to zero,
We have to solve simultaneous equation {a^2+m^5b^2=c^2+m^5d^2, a^4+m^3b^4=c^4+m^3d^4}.

a^2+m^5b^2 = c^2+m^5d^2........................................................(3)

a^4+m^3b^4 = c^4+m^3d^4........................................................(4)

Substitute a=pr+m^5qs, b=ps-qr, c=pr-m^5qs, and d=ps+qr to (3) and (4), then (4) becomes to (5),

(p^2m^2-q^2)r^2 = (p^2-q^2m^12)s^2.............................................(5)
    
Hence, we need to find the rational solution of Y^2=(p^2m^2-q^2)(p^2-q^2m^12)..(6)

Let m=2 and X=p/q,then (6) becomes to (7),
Y^2=4X^4-16385X^2+4096.........................................................(7)

Transform (7) to Weierstrass form (8),
V^2 = U^3 + U^2 - 89554944X - 325154859084.....................................(8)

By Using Mwrank(Cremona), elliptic curve (8) has rank=1 and generator is
(-3692053142663983746918803716182/707814039397310056623944449, -6412856486857860975620839587840/707814039397310056623944449).

Transform this point (U,V) of (8) to (X,Y) of (7),
we obtain (X,Y)=(69949990158596801/306781946657920, 4696288216130245645773006072054399/47057581397611436128749363200).

Then, we obtain {p,q,r,s} = {69949990158596801, 306781946657920, 5911167604843137, 12317476831120126},
{a,b,c,d} = {534407060429869176086407612538177, 859793943610761912321826231621886, 292565171139318137956759657471297, 863420822620431936290192229011966}.


Let x=n^7/(-14c^6-14d^6m+14b^6m+14a^6), then we obtain a parametric solution of (1).

x1 = n^7+1496170491324396375814393101520932926509716537937325568587201145104906701987576
     40853340924616263466922584722833269172142345169585322098113265349160098090071583631
     50397490004814847563692106657691837449096257117569191476411236352000

x2 = n^7-1496170491324396375814393101520932926509716537937325568587201145104906701987576
     40853340924616263466922584722833269172142345169585322098113265349160098090071583631
     50397490004814847563692106657691837449096257117569191476411236352000

x3 = 2n^7+24071506952305123297259170283082448675257851757388253006012444854871538990283
     78880188026589590748078670383273312891856383059767822548490448824948822341867926381
     6443911594208573410563399109734874081423063487158955177788399681536000

x4 = 2n^7-24071506952305123297259170283082448675257851757388253006012444854871538990283
     78880188026589590748078670383273312891856383059767822548490448824948822341867926381
     6443911594208573410563399109734874081423063487158955177788399681536000

y1 = n^7+8190898067398628092881925019803967782229429377271121677959182643872295577782050
     88029966362882095809744228728399419794931999624253988924714406936976271694255761642
     6859736275115645406793227234991791001861350780521991241803497472000

y2 = n^7-8190898067398628092881925019803967782229429377271121677959182643872295577782050
     88029966362882095809744228728399419794931999624253988924714406936976271694255761642
     6859736275115645406793227234991791001861350780521991241803497472000

y3 = 2n^7+241730480761350966511107889381423908888619074106901614766252095151834063495542
     892963177490799966194626784128251351769427212739893129658533169258832659546721628341
     45468112558253688726916005044040879646872239378024619747747823616000

y4 = 2n^7-241730480761350966511107889381423908888619074106901614766252095151834063495542
     892963177490799966194626784128251351769427212739893129658533169258832659546721628341
     45468112558253688726916005044040879646872239378024619747747823616000

y5 = 279968324168640071373259408572488723488588410706388315910949779841775867125507882802
     582271781665840637360813739821315933270453244992267727800469824444280894947000772779
     32532250401195703220043776000n


Accordingly, (1) has a parametric solution.

Q.E.D.　
 
 
Reference

[1].Tito Piezas:http://sites.google.com/site/tpiezas/001b