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An equation (Posted on 2014-07-03)

The equation x^xy^y=z^z has trivial integer solutions if x=1 or y=1. What is the smallest solution to x^xy^y=z^z where x, y, and z are integers and 1<x≤y?

No Solution Yet Submitted by Math Man

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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

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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