1.Introduction


We show simultaneous equation {ax^6 + by^6 + cz^6=u^6, bx^6 + cy^6 + az^6=v^6, cx^6 + ay^6 + bz^6=w^6} has a parametric solution.
     
2.Theorem
     
A simultaneous equation {ax^6 + by^6 + cz^6=u^6, bx^6 + cy^6 + az^6=v^6, cx^6 + ay^6 + bz^6=w^6} has a parametric solution.

a = (p+3q)(2p^2+3qp+3q^2)(31381059609q^44+39463780467278205p^30q^14+556647p^44+41648447526p^40q^4+20309628p^43q+371468327p^42q^2
  +4539580230p^41q^3+174940533290718p^35q^9+624712965840783p^34q^10+68951616820594110p^8q^36+2014325088999636p^33q^11+6846435089712531p^6q^38
  +1691006751659376p^5q^39+6936491484592393149p^18q^26+6732295235899706016p^19q^25+57050766369162p^3q^41+1198704501588056370p^25q^19
  +399205918728013896p^10q^34+176737799988852120p^9q^35+23436399914054442p^7q^37+4697394450752770902p^15q^29+6609757775790226050p^17q^27
  +3480659443017475407p^14q^30+2354039622839083896p^13q^31+802260435764892492p^11q^33+2839602162256628718p^23q^21+5058364384631032020p^21q^23
  +90521381789888688p^29q^15+305435102008p^39q^5+700122598717925925p^26q^18+9680033195220p^37q^7+3928427946064283325p^22q^22+43790842299402p^36q^8
  +192583181130884304p^28q^16+5913682082345601p^32q^12+345995940634164p^4q^40+380703445714310512p^27q^17+1861459283172p^38q^6+15914441357996058p^31q^13
  +648541898586pq^43+7252511554080p^2q^42+1446103121654080593p^12q^32+5810198339654342406p^16q^28+6056122504407938298p^20q^24+1911566260453685692p^24q^20)q/f

b = (-p+3q)(p+3q)(p+2q)(p+q)(2p^2+3qp+3q^2)(10460353203q^40+74319p^40+230127770466pq^39+28923178621778406p^18q^22+353176688341026p^28q^12
  +87388803928332p^4q^36+3652686215868p^32q^8+3722721620p^36q^4+6805381296377376p^8q^32+46382118477294546p^12q^28+15298539518581644p^20q^20
  +48873939361377777p^16q^24+14051955551956p^31q^9+1594194833376356p^26q^14+17175899959326p^3q^37+137622374165000p^29q^11+1113836235180903p^6q^34
  +346412891173770p^5q^35+13383808116021144p^9q^31+2985754958107110p^7q^33+34676280445206924p^11q^29+22965103019920428p^10q^30+58864952275869816p^14q^26
  +55246400357366952p^13q^27+56377088948406852p^15q^25+38831933795401404p^17q^23+11528023645943148p^21q^19+6510536522563356p^23q^17+20923956702400860p^19q^21
  +8792021618452058p^22q^18+2466706253463p^2q^38+4500832277785169p^24q^16+47043600276504p^30q^10+822703733458p^33q^7+37931375p^38q^2+26550541918p^35q^5
  +159759482675p^34q^6+427529178p^37q^3+2829816322539512p^25q^15+2319654p^39q+798057251725148p^27q^13)qp/f

c = -(-p+3q)(p+2q)(p+q)(94143178827q^44+27515314130686147p^30q^14+488089p^44+31684794232p^40q^4+16804494p^43q+295613980p^42q^2+3517959354p^41q^3
  +128628334323056p^35q^9+458182564507040p^34q^10+6468274653496074p^8q^36+1470681701924228p^33q^11+2148136067525472p^6q^38+899720548742304p^5q^39
  +1374905471475033441p^18q^26+1626742795132117530p^19q^25+75354059951478p^3q^41+638922241994057568p^25q^19+11323473307504191p^10q^34+8622954913130478p^9q^35
  +4125054146647212p^7q^37+436046831125453836p^15q^29+1049067224825196960p^17q^27+232674706614875193p^14q^30+108273641023336698p^13q^31+19457036502165144p^11q^33
  +1243277579889584760p^23q^21+1706622802219833918p^21q^23+61137292858817526p^29q^15+229355919828p^39q^5+403144361699521409p^26q^18+7160293348338p^37q^7
  +1522839520408952273p^22q^22+32273802876578p^36q^8+124722901030755826p^28q^16+4280465785571833p^32q^12+297649737811854p^4q^40+233850960996147738p^27q^17
  +1385175696851p^38q^6+11350174723146428p^31q^13+1631815099668pq^43+13790232305955p^2q^42+45108788099033061p^12q^32+717655541928098004p^16q^28
  +1746169529248988064p^20q^24+930320107918977062p^24q^20)p/f

f = (17769150p^47q+57383079901327242252p^13q^35+33663026830402735335q^36p^12+205364931198375375792q^28p^20+88745242327251362874p^14q^34
  +149947195256334638883q^26p^22+84221468182346128642q^24p^24+116012268370599437970q^25p^23+181525025833027925886q^27p^21+192952554650464068414q^31p^17
  +162006891000553524915q^32p^16+125226723032288825076q^33p^15+216577317302926722558q^29p^19+212330505730729780089q^30p^18+1548700077140832294p^31q^17
  +675427845659814003p^32q^16+332518257p^46q^2+36828594573195766059p^26q^22+3321936714593817553p^30q^18+12152967104008335p^36q^12+37033942014714492p^35q^13
  +104656428812781726p^34q^14+275258720061647892p^33q^15+6672293754443196066p^29q^19+12562123988562983472p^28q^20+22193387687732782410p^27q^21+488089p^48
  +110707208174126295q^42p^6+409993369452364950p^7q^41+8413148167409958678p^10q^38+3523050186470936208p^9q^39+1292494642766534673p^8q^40+17797249193786411112p^11q^37
  +11859905443590p^41q^7+320696054262p^43q^5+2102881886563p^42q^6+257919286391424p^39q^9+1021617438503946p^38q^10+3680022859106112p^37q^11+57442721116650921666p^25q^23
  +4233211434p^45q^3+40998873801p^44q^4+58634497593741p^40q^8+677935491086430q^45p^3+4609532464906401p^4q^44+24980585664717162p^5q^43+76726690744005p^2q^46+6213449802582pq^47+282429536481q^48)
  
[x,y,z]=[2p^4+4p^3q-5p^2q^2-12pq^3-9q^4, 3p^4+9p^3q+18p^2q^2+21pq^3+9q^4, -p^4-10p^3q-17p^2q^2-12pq^3]
[u,v,w]=[p^4-3p^3q-14p^2q^2-15pq^3-9q^4, 3p^4+8p^3q+9p^2q^2, 2p^4+12p^3q+19p^2q^2+18pq^3+9q^4]
   
p,q are arbitrary.

 
Proof.

ax^6 + by^6 + cz^6.................................................................................(1)

bx^6 + cy^6 + az^6.................................................................................(2)

cx^6 + ay^6 + bz^6.................................................................................(3)

We obtain the solution of equation (1),(2), and (3) for {a,b,c} below.

a = (-y^6x^6v^6+w^6y^12+z^12v^6-y^6z^6u^6+x^12u^6-x^6z^6w^6)/(-3y^6z^6x^6+x^18+y^18+z^18)..........(4)

b = (x^12v^6-x^6w^6y^6-x^6z^6u^6-z^6v^6y^6+z^12w^6+u^6y^12)/(-3y^6z^6x^6+x^18+y^18+z^18)...........(5)

c = (v^6y^12-y^6x^6u^6-y^6z^6w^6+w^6x^12-z^6v^6x^6+z^12u^6)/(-3y^6z^6x^6+x^18+y^18+z^18)...........(6)

Let x^6+y^6+z^6=u^6+v^6+w^6 and obtain one of parametric solutions of x^6+y^6+z^6=u^6+v^6+w^6 below.
See A collection of the parametric solutions for equal sums of sixth powers  

[x,y,z]=[2p^4+4p^3q-5p^2q^2-12pq^3-9q^4, 3p^4+9p^3q+18p^2q^2+21pq^3+9q^4, -p^4-10p^3q-17p^2q^2-12pq^3]
[u,v,w]=[p^4-3p^3q-14p^2q^2-15pq^3-9q^4, 3p^4+8p^3q+9p^2q^2, 2p^4+12p^3q+19p^2q^2+18pq^3+9q^4].....(7)
p,q are arbitrary.

Substitute the parametric solution (7) to equation (4),(5), and (6).

Hence we obtain a parametric solution of simultaneous equation.


Q.E.D.


3.Example

Let [p,q]=[1,2].
We obtain
[x,y,z]=[50, 81, 37]
[u,v,w]=[65, 11, 78]
[a,b,c]=[1216403176547067112456/1527336301036554474937, 340829850234815072568/1527336301036554474937, -29896725745327710087/1527336301036554474937].