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Warning: I am an amateur recreational mathematician in highschool who knows everything I know about Fermat's Last Theorem by watching Numberphile videos on YouTube, so please forgive me if I'm getting anything wrong here.

Fermat's Last Theorem, to my understanding, states that $a^n+b^n \neq c^n$ where $n \gt 2$.
Could you have $a^n+b^n=c^n$ for $1< n<2$, like a solution(s) for $n=1.2$ or $n=\pi/2$?
If so, are there any known examples?

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Fermat's Last Theorem states that $a^n+b^n\ne c^n$ where $a,b,c$ are non-zero integers and $n$ is an integer $>2$.

If we drop these restrictions, note for example, that $15^2+26^2<30^2 $ but $15^1+26^1>30^1$, so that by continuity, there exists $x$ between$1$ and $2$ with $15^x+26^x=30^x$. There is virtually nothing special about the numbers $15, 26, 30$ involved in this.

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  • $\begingroup$ @DietrichBurde: there are some non-integer $n$ for which $a^n+b^n = c^n$ has positive integer solutions $a,b,c$. And for each positive $n$ (integer or not), there are positive non-integer solutions $a,b,c$ $\endgroup$
    – Henry
    May 15, 2017 at 15:37
  • $\begingroup$ @Dietrich Burde, No. $\endgroup$
    – marshal craft
    May 15, 2017 at 15:39
  • $\begingroup$ Aside from guess-and-check methods, how would one calculate the value of x in your example? $\endgroup$
    – Tom Dacre
    May 15, 2017 at 15:45
  • $\begingroup$ @TomDacre so far, I don't think a general method has been found to solve equations as $a^x+b^x=c^x$ for $x$, so we probably can only approximate $x$ rather than finding an exact expression for it $\endgroup$
    – vrugtehagel
    May 15, 2017 at 15:51
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The equation $$2^n+3^n=4^n$$ has a root between $1$ and $2$.

Its decimal expansion is given by:

1.507126591638653133986883360838631164373994094485656896675364359443814733804851572592281309243976770528554306625049366654408029194186594013522018895286000020146247128113782170813211284377777773600060489310566796113814627660108720626522220627977067752672882746839209662298060663426982416719874695750697238147739634291096518542479744540255890702401981177227291454008416333371393809300177967326360567605679934582837172426237968324627726208465884390666931502322651023788749176926837939881443328147919480889369825352941087796490585245687788847387672387232430458190906513596628525127764075077811064894130853212657381337891178124233424703245206444802843096081095731264484064758369062116723942729841729235878222504282203342331331248338316510500122223224450504372485567136385983450419132770012239575299164493610802858175306919659548808746923830804415380800641753609793059895055090888799023785252722966839035456248484394864133608969878080761690637877515735668292611101309732761547732528421265014933808228311143953950639358358715261288625830486035158241925457181405428084480649662197325883514793401278553668138911255170936781572720446836722907903165987057239870786644614242341455954403263074351574135304971673420607843916726557015096554441952800282027197706749799382088470819744261396406103335478105323897728872698737902550734334670845347465645169163418493117969978061879954234386222166579685393688743538101803283729103916443679535520374123153976976670144693979873076055599145449671549460960096932869251280640690200875622214849650938211541732055183620660750966881808669418538373273091800919251902296081993244822974424131569511785991566902415220614635976234809192276527585277773943145644845358284364640245980904854682250074830046993945629257630223834035019795463186907886713777542983409887158054291002096381662248264087645915196337857779289865835102582253003341360446194346604426035891439219605143053833994542967060006556350892087273529090674942054012773824417730952353803918321312707513980628942581426985002559802541230157909129657441089493976226326602911338696342477259509114677832793485902033437517348278581131045070266373936920689216466140151698550998946222448466119648409269985614134231368222785331465048367068828656436438523908379260557485342912687908573446102427621360877245014621076890899946908631037273555852653656392084681502381776847509754579928585101063093471809312869587674492076423700333573433330452865523547268417886693456887379388410057862814837105291679333851252026543044049057071295828900473213845254567278695108829813807147203251761900060630880130349976...

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    $\begingroup$ If I may ask, how was that value calculated? $\endgroup$
    – Tom Dacre
    May 15, 2017 at 15:42
  • $\begingroup$ @TomDacre WolframAlpha. $\endgroup$
    – Kenny Lau
    May 15, 2017 at 15:43
  • $\begingroup$ If you don't like this answer then please never visit ppcg, your head will explode $\endgroup$
    – Darren H
    May 15, 2017 at 17:50

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