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In other words, $x(x+1)(2x+1)=6y^2$ has no nontrivial integral solutions. I thought this is a well-known result, but surprisingly could not find a recorded (easy) proof.

Can someone provide a proof here?

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Counterexamples: ($x=1$, $\;$ $y=1$) ($x=24$, $\;$ $y=70$)

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  • $\begingroup$ OK, thanks. But is there any proof that there are only two solutions, which I think is true. $\endgroup$ – Liam_math Feb 12 '18 at 21:37
  • $\begingroup$ @Liam_math yes, just not easy. However, short, which is good too. See my answer. $\endgroup$ – Will Jagy Feb 12 '18 at 22:30
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This is Lucas' problem of the square pyramid. A proof with elliptic curves was given by G. N. Watson in 1919. An "easier" proof was given in 1952 by W. Ljungren. I found this out in Mordell's book, he does not reproduce either proof.

There is a proof on the easier side by Anglin, 1990.

If you look up "Lucas square pyramid" you can get a link to a paper by Bennett from Acta Arithmetica, 2002.

There is no truly easy proof of this, there are just degrees of background required. Some of the proofs are called "elementary." My reaction is that elementary is in the eye of the beholder.

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