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An equation (Posted on 2014-07-03) Difficulty: 3 of 5
The equation xxyy=zz has trivial integer solutions if x=1 or y=1. What is the smallest solution to xxyy=zz where x, y, and z are integers and 1<x≤y?

No Solution Yet Submitted by Math Man    
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Some Thoughts computer exploration | Comment 1 of 5
   10   for Tot=3 to 1000
   20     for X=2 to int(Tot/2)
   30        Xp=X^X
   40        Y=Tot-X
   50        Yp=Y^Y
   60        Prod=Xp*Yp
   70        Low=2:High=2*Y
   72        while log(High)*High<=log(Prod)
   73           inc High
   74        wend
   85        repeat
   90          M=int((Low+High)/2)
   95          M2=M^M
  100          if M2>Prod then High=M-1
  110          if M2<Prod then Low=M+1
  120        until Low>High or M2=Prod
  130        Z=M
  210        Z=int(Z+0.5)
  220        if Z^Z=Prod then print X;Y;Z,X^X;Y^Y,Prod,Z^Z:inc Solct
  225        if Solct>30 then end
  230     next X
  240   next Tot
  
does not find any solutions up to the point where the total of x and y reaches 595 and the product of x^x * y^y reaches

 1983978175966114206105992795156965609830078354477006722055266903216207615848029
70995127753831672467947056009777276939232774901947206435565141972385217470939481
58230540485333531628612973217479276245706045866659680030506543343377953660813355
98908717311537773431730447450224917774560173379960741291730503940290267402949770
17015036642965390235605716431064138034607205905571246810274527919092089919515152
91200628496356429580061439275187419123643114364914113537174420360873691875221452
88315895330644728410273603286239351660712838121526566104825966889641346086117438
37700106539029768716299516367993120638172780125849247927191324126028678811950797
21927924877302487655042933245733145167133551069001073090763852912657043050641179
80576476464324968601241075757564647504969094963804249073889516408965602009667893
06425635410208577871652184935516292157826517111059626532788754192169720012614988
27613417925565957220857225669488301292064803996390565306121012136732250199355731
05962373530010533152443757459033975758449360627666467252351862946890846389142156
06331281250000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000

where x=5 and y=590.

All possible totals of x and y up through 594 were searched for solutions, using a binary search through z values that might fit.


  Posted by Charlie on 2014-07-04 11:51:38
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