Reference request: Proof of the non-existence of three consecutive perfect powers
问题内容
I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers.
While reading Wacław Sierpiński's 250 Problems in Elementary Number Theory, I came across a remark stating that A. Mąkowski proved this result, with the author noting that the proof is "difficult" and citing M. N. Khatri (Colloq. Math., 9 (1962), 297). I have been unable to locate the text of the proof itself.
Could you point me toward a source where this specific derivation is presented? I am interested in the methodology used, particularly given that the text describes the proof as non-trivial.
回答 (1)
This was proved by Makowsky, Three consecutive integers cannot be powers.
Makowsky shows that the system of equations $$x^p-y^q=1,\quad y^q-z^r=1$$ (for $x,y,z$ whole numbers and $p,q,r$ prime) has no solutions, which gives the desired result. A result of Cassels (On the equation ...) gives $q\vert x$ and $q\vert z$, so $q=2$; we then get $x^p=y^2+1$, which was earlier proved by Lebesgue (Sur l'impossibilite ...) to have no integer solutions.
It's amusing to note that changing the value of $3$ in either direction changes the character of the question immensely: when $3=2$ we get Catalan's conjecture/Mihaelescu's theorem, and when $3=4$ we get an easy number theory problem (and it's from a deleted answer to that question that I found Makowsky's paper, incidentally).