Prove that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$

Question

Fermat claimed that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$, but did not write a proof.
Who can provide a proof that a high school student can accept?

Thank you for your help An answer given by the Chinese friends: similar to the integer division algorithm, but the Chinese, in front of first give some basic properties of the final is proved.Please look at. enter image description here

You should be careful to mention you want integer solutions. — Steven-Owen, May 24 '12 at 07:33
Per chance the unique factorization in the extension $Q((-2)^{1/2})$ is acceptable? — awllower, May 24 '12 at 07:36
A solution using the fact that $\mathbb Z[\sqrt{-2}]$ is an UFD is given in the book Titu Andreescu,Dorin Andrica,Ion Cucurezeanu: An Introduction To Diophantine Equations, p.169. However, this is not accessible to high school students. — Martin Sleziak, May 24 '12 at 07:38
Ah! Infinite descent is well-known already to high school students, right? If not, then why not explain to them? If so, then an approach might be acceptable, as shown in the book Number Theory: An approach through history, from Hammurapi to Legendre by *A.Weil*. If I recall correctly. — awllower, May 24 '12 at 07:39
@Martin: the use of ${\mathbf Z}[\sqrt{-2}]$ is not acceptable to high school students? Gosh, I learned that proof when I was in high school... :) — KCd, May 24 '12 at 08:31
@awllower I found solution of $y^2=x^3-2x$ using infinite descent in Weil's book, p.150. Is the solution of $y^2=x^3-2$ given there too? — Martin Sleziak, May 24 '12 at 11:19
I don't have the book handy, but I think this equation is solved in Uspensky & Heaslett, which is an introductory text in Number Theory. It is, by the way, an example of a "Mordell equation", and a websearch on that term will give you some idea of the difficulties involved. (Aside to @KCd, bet you learned it same place I did, in Columbus.) — Gerry Myerson, May 24 '12 at 12:11
This question also deals with the same equation: How to find all rational points on the elliptic curves like $y^2=x^3-2$. — Martin Sleziak, May 24 '12 at 12:52
I checked; U & H, Elementary Number Theory, pages 398 to 399. The proof is indeed via ${\bf Z}[\sqrt{-2}]$, but oddly never explicitly mentions use of unique factorization. — Gerry Myerson, May 25 '12 at 01:47
@MartinSleziak: Allow me, please, to put my hand on the book again, so as to verify my previous assertion. Apology for any inconvenience. — awllower, May 27 '12 at 07:42

score 4 · Answer 1 · answered May 27 '12 at 10:49

About the only proof of this result I have ever seen is the one using unique factorization in the quadratic domain $Z[\sqrt{-2}]$. Using infinite descent it is possible to determine all rational points on the elliptic curve, and showing that $(3,5)$ is the only integral point seems to require stuff like Baker's theorem. I have been looking for a proof that Fermat could have understood for years, and would be grateful if anyone could come up with one.

https://math.stackexchange.com/a/4038301/93271 – Kieren MacMillan Aug 10 '21 at 14:40 — Kieren MacMillan, Aug 10 '21 at 14:40

DonAntonio · Answer 2 · 2012-05-25T12:57:59.867

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The following paper comes as close as I could find to be self-contained and ""basic"" in its proof. Please do note they prove there that $\,(5,3)\,$ is the only integer solution of the diophantine eq. $\,y^3-x^2=2\,$ , and that they use the notation $\,x\wedge y$ to denote the gcd of two integers $\,x\,,\,y\,$

Added: Oops, sorry! Didn't notice I didn't write down the link. Here it is http://www.normalesup.org/~baglio/maths/26number.pdf

Please notice the paper seems to be written by advanced H.S. students and/or beginning university ones, and the language is rather sloppy.

edited May 25 '12 at 12:57

answered May 24 '12 at 11:49

DonAntonio

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We're waiting... – TonyK May 24 '12 at 11:51
Waiting I would like to Fermat Is deceiving you what? Solve elementary mathematics are very interested in mathematics master, I know there are Euler website http://www.eulersociety.org/ There are other master site? – tianzhidaosunyouyu May 25 '12 at 06:07
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But, as far as I perceive, this still avails of the unique factorising property to resolve the problem, while showing no attempt to evident this assumption. Well, t'is what being elementary means? Assume something without proof? Might I agree not? – awllower May 27 '12 at 07:37

score 1 · Answer 3 · answered Aug 10 '21 at 14:42

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There is a completely elementary solution — accessible to any reasonably advanced high school math student — due essentially to Stan Dolan. See this answer to a duplicate question.

answered Aug 10 '21 at 14:42

Kieren MacMillan

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Prove that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$

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