Talk:Proof of Fermat's Last Theorem for specific exponents

Just wondering, who would stumble across this page without knowing what even and odd numbers are. It's a fairly high-level mathematics article. — Preceding unsigned comment added by 82.6.96.22 (talk • contribs) 21:18, 18 February 2011

I would have thought that too, if I had not had years of experience of teaching mathematics. It's amazing how many people, for example, for some reason don't think that 0 is even. Also, many people have encountered the concepts "even" and "odd" only in the context of positive integers, and unless they are explicitly told that negative integers are also odd or even, are very likely to take "even", for example, as meaning "even and positive". This applies even to many people of quite high mathematical ability, if they have never encountered the sue of the words "odd" and "even" in any context other than the positive integers. The editor who uses the pseudonym "JamesBWatson" (talk) 14:31, 28 February 2014 (UTC)

This is a Wikipedia talk page, not a forum

A Wikipedia talk page is for discussions relating to editing the corresponding Wikipedia article, not for publishing supposed proofs of theorems, for linking to such supposed proofs on other web sites, or anything similar. An editor from Vietnam has recently been using several talk pages of articles to publicise attempts at mathematical proofs. To that editor, please don't. There are plenty of internet forums, blogs, etc where the kind of thing you have been doing would be accepted, but a Wikipedia talk page is not one of them. Continuing in the same way may lead to the range of IP addresses that you use being blocked from editing. That would not be a disaster, since the majority of the other editing from that range is vandalism anyway, but it would be a pity, since there have occasionally been constructive edits from that range, and they would be blocked along with the unconstructive edits. The editor who uses the pseudonym "JamesBWatson" (talk) 14:22, 28 February 2014 (UTC)

n=4, case A

I don't think case A actually uses the parity of x or y. So could the whole proof just be replaced with case A?  AltoStev (talk) 23:25, 25 October 2022 (UTC)

The parametrization of the Pythagorean triple explicitly uses that we know which leg is even (namely, z). 66.44.22.126 (talk) 01:15, 26 October 2022 (UTC)

Hmm, in that case could a non-parity dependent Pythagorean triple proof this be used? (Note: I don't know how to put multiple equations in <math>)

a² + b² = c²

c² − a² = b²

(c + a)(c − a) = b²

(c + a) ∕ b = b ∕ (c − a)

Let m ∕ n be (c + a) ∕ b.

Since there’s no difference between a² and (-a)², let’s assume that a, b, c > 0.

Thus m and n are integers > 0.

Solving for c ∕ b and a ∕ b we get:

m ∕ n = (c + a) ∕ b

c ∕ b + a ∕ b = m ∕ n

m ∕ n = b ∕ (c − a)

n ∕ m = (c − a) ∕ b

c ∕ b − a ∕ b = n ∕ m

-c ∕ b + a ∕ b = -n ∕ m

c ∕ b = (m ∕ n + n ∕ m) ∕ 2 = (m² + n²) ∕ 2mn

a ∕ b = (m ∕ n − n ∕ m) ∕ 2 = (m² − n²) ∕ 2mn

Equating the fractions, one gets

a = (m² + n²), b = 2mn, c = (m² − n²)

where 0 < {n, m} < a  AltoStev (talk) 01:10, 28 October 2022 (UTC)

Well obviously this is not independent of parity, since you've derived that the (nominally arbitrary) leg b is always even. (I haven't tried to find where exactly this happened in your derivation.) --66.44.22.126 (talk) 02:53, 31 October 2022 (UTC)

Ohhh hmm. I think I somehow proved (if primitive pythagorean triple then one of the legs is even).

Copying from https://en.wikipedia.org/wiki/Pythagorean_triple#Proof_of_Euclid's_formula btw.

Say the legs were odd. Then a² and b² are both 1 mod 4, so a² + b² is 2 mod 4. But c² can't be 2 mod 4, so therefore at least one leg is even.  AltoStev (talk) 21:58, 6 November 2022 (UTC)

Ok yeah, the parameterization of the pythagorean triple does NOT require we know which leg is even: https://us.metamath.org/mpeuni/pythagtrip.html  AltoStev (talk) 16:33, 20 August 2024 (UTC)

Note: We don't have the coprime requirement on either the parameterization, which is ok since we don't have the coprime requirement on the triple either, so we can infinite descent on counterexamples in general (for NN) and expand to ZZ  AltoStev (talk) 16:40, 20 August 2024 (UTC)

Nevermind, the link has `k` which factors everything, so it needs to be primitive after all  AltoStev (talk) 16:59, 20 August 2024 (UTC)