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I am wanting to start a proof for the following statement:

Prove that there is an integer $n$ such that $5^n=n^5$.

Would proof by construction be an appropriate method to use?

Then would proof by induction be an appropriate method to use?

user430574
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    Let $n=5$ . . . – Kenta S Jun 18 '20 at 04:43
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    Just show the value $n.$ You can prove $n$ is unique using unique factorization. – Thomas Andrews Jun 18 '20 at 04:46
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    Proof "by inspection". – Blue Jun 18 '20 at 04:47
  • @Blue What is "by inspection"? I have only learned Construction, Contradiction, Induction so far. – user430574 Jun 18 '20 at 04:49
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    Construction means you just construct a solution. By inspection means you looked at it and can see that $n = 5$ works. It was still a constructed solution, though. – Derek Luna Jun 18 '20 at 04:58
  • @user430574: click on the blue words in Blue’s comment for an explanation of “by inspection” – J. W. Tanner Jun 18 '20 at 04:59
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    @user430574: The link goes to a Wikipedia entry describing the phrase "by inspection". Basically, it means "proof by look!-it-works" when there's an "obvious" solution. As @ Derek notes, this probably counts as a proof by construction; there just isn't much construction effort to make. – Blue Jun 18 '20 at 05:00
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    @user430574, perhaps a more worthwhile aim would be to prove that there is only one integer $n$ satisfying your equation? – Auslander Jun 18 '20 at 05:17
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    Induction proofs are about properties that hold for all natural numbers or all natural numbers beginning with some specific number, this cannot be applied here. – Peter Jun 18 '20 at 07:36

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