As many of you know, I have "published" a few tachyon papers on vixra. The first were naïve but I’ve been learning with each iteration.

This past April, I finally felt that I had graduated to a higher school and submitted a paper to AJP. It was rejected but one reviewer said:

“While the topic is interesting, the article is not self-contained. The terminology used is not sufficiently well defined and not physically motivated. This situation would place heavy demands on the typical AJP reader.
"Furthermore, the speculative nature of the article suggests to me that AJP is not the appropriate journal for this article.
"However, I would encourage the author to reformulate the presentation of these ideas with Minkowski spacetime geometry.”

This all old news, but I did what the reviewer suggested and and submitted it to a less-reputable “Progress in Physics." It was rejected fairly quickly in October. I don’t think it even made it to reviewers, the editor done me in. I’ve made a few typo corrections and have published it on Vixra:

https://vixra.org/abs/2112.0147

I narrowed the focus on my third attempt and aimed at AJP again. I felt that one method used in textbooks and lecture notes to “disprove” FTL phenomena was incorrect and pointed out it shouldn’t be used (Method I for those familiar with my nomenclature). It was a long time in review and the reviewer finally said:

“ If the journal has nothing better submitted, I don't see any serious harm in publishing the paper.”

Apparently, the editors thought they did :-|

The reviewer also claimed that a signal having zero energy could still be detected (because the 4-momentum is NOT zero).

I think he's wrong, but at least the argument has gotten down to the nuts-and-bolts level. I just have to prove that he's wrong (or prove that he's right).

Gary

On Thursday, December 30, 2021 at 7:36:52 AM UTC-8, hit...@yahoo.com wrote:
> As many of you know, I have "published" a few tachyon papers on vixra. The first were naïve but I’ve been learning with each iteration.
>
> This past April, I finally felt that I had graduated to a higher school and submitted a paper to AJP. It was rejected but one reviewer said:
>
> “While the topic is interesting, the article is not self-contained. The terminology used is not sufficiently well defined and not physically motivated. This situation would place heavy demands on the typical AJP reader.
>
> "Furthermore, the speculative nature of the article suggests to me that AJP is not the appropriate journal for this article.
>
> "However, I would encourage the author to reformulate the presentation of these ideas with Minkowski spacetime geometry.”
>
> This all old news, but I did what the reviewer suggested and and submitted it to a less-reputable “Progress in Physics." It was rejected fairly quickly in October. I don’t think it even made it to reviewers, the editor done me in. I’ve made a few typo corrections and have published it on Vixra:
>
> https://vixra.org/abs/2112.0147
>
> I narrowed the focus on my third attempt and aimed at AJP again. I felt that one method used in textbooks and lecture notes to “disprove” FTL phenomena was incorrect and pointed out it shouldn’t be used (Method I for those familiar with my nomenclature). It was a long time in review and the reviewer finally said:
>
> “ If the journal has nothing better submitted, I don't see any serious harm in publishing the paper.”
>
> Apparently, the editors thought they did :-|
>
> The reviewer also claimed that a signal having zero energy could still be detected (because the 4-momentum is NOT zero).
>
> I think he's wrong, but at least the argument has gotten down to the nuts-and-bolts level. I just have to prove that he's wrong (or prove that he's right).
>
> Gary

Gary,

Just keep up the imbecilities, eventually some rag will publish them.

On Thursday, December 30, 2021 at 7:43:12 AM UTC-8, Dono. wrote:
> On Thursday, December 30, 2021 at 7:36:52 AM UTC-8, hit...@yahoo.com wrote:
> > As many of you know, I have "published" a few tachyon papers on vixra. The first were naïve but I’ve been learning with each iteration.
> >
> > This past April, I finally felt that I had graduated to a higher school and submitted a paper to AJP. It was rejected but one reviewer said:
> >
> > “While the topic is interesting, the article is not self-contained. The terminology used is not sufficiently well defined and not physically motivated. This situation would place heavy demands on the typical AJP reader.
> >
> > "Furthermore, the speculative nature of the article suggests to me that AJP is not the appropriate journal for this article.
> >
> > "However, I would encourage the author to reformulate the presentation of these ideas with Minkowski spacetime geometry.”
> >
> > This all old news, but I did what the reviewer suggested and and submitted it to a less-reputable “Progress in Physics." It was rejected fairly quickly in October. I don’t think it even made it to reviewers, the editor done me in. I’ve made a few typo corrections and have published it on Vixra:
> >
> > https://vixra.org/abs/2112.0147
> >
> > I narrowed the focus on my third attempt and aimed at AJP again. I felt that one method used in textbooks and lecture notes to “disprove” FTL phenomena was incorrect and pointed out it shouldn’t be used (Method I for those familiar with my nomenclature). It was a long time in review and the reviewer finally said:
> >
> > “ If the journal has nothing better submitted, I don't see any serious harm in publishing the paper.”
> >
> > Apparently, the editors thought they did :-|
> >
> > The reviewer also claimed that a signal having zero energy could still be detected (because the 4-momentum is NOT zero).
> >
> > I think he's wrong, but at least the argument has gotten down to the nuts-and-bolts level. I just have to prove that he's wrong (or prove that he's right).
> >
> > Gary
> Gary,
>
>
> Just keep up the imbecilities, eventually some rag will publish them.

For example, if you pay 500$, this rag will publish anything: https://www.academia.edu/letters/about

Le 30/12/2021 à 16:36, Gary Harnagel a écrit :
> As many of you know, I have "published" a few tachyon papers on vixra. The
> first were naïve but I’ve been learning with each iteration.
>
> This past April, I finally felt that I had graduated to a higher school and
> submitted a paper to AJP. It was rejected but one reviewer said:
>
> “While the topic is interesting, the article is not self-contained. The
> terminology used is not sufficiently well defined and not physically motivated.
> This situation would place heavy demands on the typical AJP reader.
>
> "Furthermore, the speculative nature of the article suggests to me that AJP is
> not the appropriate journal for this article.
>
> "However, I would encourage the author to reformulate the presentation of these
> ideas with Minkowski spacetime geometry.”
>
> This all old news, but I did what the reviewer suggested and and submitted it to
> a less-reputable “Progress in Physics." It was rejected fairly quickly in
> October. I don’t think it even made it to reviewers, the editor done me in.
> I’ve made a few typo corrections and have published it on Vixra:
>
> https://vixra.org/abs/2112.0147
>
> I narrowed the focus on my third attempt and aimed at AJP again. I felt that
> one method used in textbooks and lecture notes to “disprove” FTL phenomena was
> incorrect and pointed out it shouldn’t be used (Method I for those familiar with
> my nomenclature). It was a long time in review and the reviewer finally said:
>
> “ If the journal has nothing better submitted, I don't see any serious harm in
> publishing the paper.”
>
> Apparently, the editors thought they did :-|
>
> The reviewer also claimed that a signal having zero energy could still be
> detected (because the 4-momentum is NOT zero).
>
> I think he's wrong, but at least the argument has gotten down to the
> nuts-and-bolts level. I just have to prove that he's wrong (or prove that he's
> right).
>
> Gary

I am amazed that scientists can do serious research on tackyons.
I worked for forty years on the theory of relativity (but not only that)
and I realized that the fact that one cannot find anything faster than
light was not due to a property of particles, but to a property of space
itself.
Consider the following idea: there is no such thing as a natural number
square that is double another natural number square.
16 is a square, but it is not double 9.
100 is a square, but it is not twice 25 or 36.
If you put the problem on paper, you will quickly understand why no square
is the double of another, nor why no cube is the triple of another.
And you will understand that the fault does not lie with the thousand
numbers that you will choose to make attempts, but simply with the fact
that sqrt2 and sqrt3 are not natural numbers.
Do you see what I mean, and why I always said tackyons couldn't exist?
This does not mean that I am not sure that we can make instantaneous
transport of information but under certain conditions (quantum effects)
and while knowing that this does not harm the obvious notion of causality.

R.H.

Richard Hachel wrote:

>> I think he's wrong, but at least the argument has gotten down to the
>> nuts-and-bolts level. I just have to prove that he's wrong (or prove
>> that he's right). Gary
>
> I am amazed that scientists can do serious research on tackyons.
> I worked for forty years on the theory of relativity (but not only that)
> and I realized that the fact that one cannot find anything faster than
> light was not due to a property of particles, but to a property of space
> itself.

both wrong. It's deeper than that. About the characteristics of the
spacetime manifold, the world you inhabit. It's a "take it or leave it",
built in. Without which you have no world to begin with. Based on a few
constants, which since you insist, may vary a little, but overall these
are tight together.

On Thursday, December 30, 2021 at 8:48:39 AM UTC-7, Dono. wrote:
>
> On Thursday, December 30, 2021 at 7:43:12 AM UTC-8, Dono. wrote:
> >
> > On Thursday, December 30, 2021 at 7:36:52 AM UTC-8, hit...@yahoo.com wrote:
> > >
> > > As many of you know, I have "published" a few tachyon papers on vixra.. The first were naïve but I’ve been
> > > learning with each iteration.
> > >
> > > This past April, I finally felt that I had graduated to a higher school and submitted a paper to AJP. It was
> > > rejected but one reviewer said:
> > >
> > > “While the topic is interesting, the article is not self-contained. The terminology used is not sufficiently well
> > > defined and not physically motivated. This situation would place heavy demands on the typical AJP reader.
> > >
> > > "Furthermore, the speculative nature of the article suggests to me that AJP is not the appropriate journal for
> > > this article.
> > >
> > > "However, I would encourage the author to reformulate the presentation of these ideas with Minkowski
> > > spacetime geometry.”
> > >
> > > This all old news, but I did what the reviewer suggested and and submitted it to a less-reputable “Progress in
> > > Physics." It was rejected fairly quickly in October. I don’t think it even made it to reviewers, the editor done me
> > > in. I’ve made a few typo corrections and have published it on Vixra:
> > >
> > > https://vixra.org/abs/2112.0147
> > >
> > > I narrowed the focus on my third attempt and aimed at AJP again. I felt that one method used in textbooks
> > > and lecture notes to “disprove” FTL phenomena was incorrect and pointed out it shouldn’t be used (Method
> > > I for those familiar with my nomenclature). It was a long time in review and the reviewer finally said:
> > >
> > > “ If the journal has nothing better submitted, I don't see any serious harm in publishing the paper.”
> > >
> > > Apparently, the editors thought they did :-|
> > >
> > > The reviewer also claimed that a signal having zero energy could still be detected (because the 4-momentum
> > > is NOT zero).
> > >
> > > I think he's wrong, but at least the argument has gotten down to the nuts-and-bolts level. I just have to prove
> > > that he's wrong (or prove that he's right).
> > >
> > > Gary
> >
> > Gary,
> >
> > Just keep up the imbecilities, eventually some rag will publish them.

You still don't understand. Didn't you even comprehend what the reviewers actually wrote? Go back
and reread them until you DO understand.

It IS down to nuts-and-bolts because it IS something that can be determined theoretically.

> For example, if you pay 500$, this rag will publish anything: https://www..academia.edu/letters/about

You claimed that AJP often publishes nonsense when I submitted the first paper. Remember?
You seem to be wrong A LOT! But don't worry, being wrong is educational for those with awareness.
No help to those who aren't, though.

“To hate being wrong is to change your opinion when you are
proven wrong; whereas pride, even when proven wrong, decides
to go on being wrong.” ― Criss Jami

Tachyons haven't been "proven" wrong, but there are many with the opinion that they don't exist and a
few with the opinion that they do. Fortunately, science isn't about taking a vote, but there are many
whose pride causes them to be close-minded and believe their opinions are fact.

I'm awaiting the conclusive KATRIN results, but the preliminary -1 eV² likely value sure looks like
tachyon in big bright lights to me! 95% probability that the electron antineutrino has imaginary mass.
Only 5% that it is a bradyon or luxon.

Now don't pull that baloney that the m² measurements are the difference between the neutrino
flavors. You were clearly dead wrong in asserting that nonsense. Just one more example of you
being wrong.

Gary

On Thursday, December 30, 2021 at 11:26:03 AM UTC-8, hit...@yahoo.com wrote:

> You still don't understand.

I understand, I pointed out the imbecilities in the many revisions of your "papers" repeatedly. Yet, you insist. Which is good. For entertainment purposes.

On Thursday, December 30, 2021 at 11:26:03 AM UTC-8, hit...@yahoo.com wrote:

> Now don't pull that baloney that the m² measurements are the difference between the neutrino flavors.

They are. But you are to dumb to learn.

Atom's can't reach the speed of light by math.
There is no bridging finite math to infinite
as it would go on forever without completing.

If tachyons FTL come backward in time
with their negative energy we would see
their energy annihlation with positive
material they would encounter.

Le 30/12/2021 à 23:22, "mitchr...@gmail.com" a écrit :
> Atom's can't reach the speed of light by math.
> There is no bridging finite math to infinite
> as it would go on forever without completing.
>
> If tachyons FTL come backward in time
> with their negative energy we would see
> their energy annihlation with positive
> material they would encounter.

The problem is not there.
Problem is, it's just absurd.
How absurd is the notion of reciprocal dilation of times without going
through Richard Hachel.
It is sad to see such a beautiful theory so misunderstood.
You can't go faster than light, not because you can't find small particles
strong enough to do it, but simply because the idea "in itself" is absurd.
It's like I said, looking indefinitely for two squares of integers, one of
which is double the other (or triple the other). We can easily prove that
a² = 2b² or that a² = 3b² would have to be.
Someone who does not know mathematics will then exhaust himself looking
for a matching a and a b without ever finding it, and the search will be
endless. His belief in a possible outcome is a total delusion.
The same goes for c. We cannot exceed this speed, because it is impassable
by its very nature.
Vo = Vr / (1 + Vr² / c²)
Vo will never be able to reach c.
Let Vr = 10c; or 1000c; or 100,000c.
Nothing will ever do.

R.H.

On Thursday, December 30, 2021 at 3:02:10 PM UTC-8, Richard Hachel wrote:
> Le 30/12/2021 à 23:22, "mitchr...@gmail.com" a écrit :
> > Atom's can't reach the speed of light by math.
> > There is no bridging finite math to infinite
> > as it would go on forever without completing.
> >
> > If tachyons FTL come backward in time
> > with their negative energy we would see
> > their energy annihlation with positive
> > material they would encounter.
> The problem is not there.

Sure it is. You just don't want the problem that is in math manifestation.

> Problem is, it's just absurd.

The math acceptance is...

Mitchell Raemsch

On Thursday, December 30, 2021 at 12:34:38 PM UTC-7, Dono. wrote:
>
> On Thursday, December 30, 2021 at 11:26:03 AM UTC-8, hit...@yahoo.com wrote:
> >
> > Now don't pull that baloney that the m² measurements are the difference between the neutrino
> > flavors.
>
> They are.

Wongo, DON'tknOw. YOU are the only fool promulgating this pernicious poop. Seems like YOU:

> are to dumb to learn.

> I understand,

Obviously, you don't :-))

> I pointed out the imbecilities in the many revisions of your "papers" repeatedly.

So why did the reviewers not agree with you? Just because they recommended that the
papers not be published doesn't mean they're "imbecilities." Your imbecilic personal
attacks seem to be a case of low self esteem.

This is clear from your claims that you have "pointed out" problems when, in fact, you
did nothing of the sort. Your "points" were inane bull poop, just like your claim about
neutrino mass measurements. There's no point in continuing talking to someone that
has no critical thinking associated with their repeated attempts at character
assassination. You spew hatefulness with every post.

On Thursday, December 30, 2021 at 3:48:33 PM UTC-8, hit...@yahoo.com wrote:

> > I pointed out the imbecilities in the many revisions of your "papers" repeatedly.
> So why did the reviewers not agree with you?

They did, all of them rejected your "paper". No exception. This is why your garbage is relegated to the vixra cesspool. Where it belongs.

> Just because they recommended that the
> papers not be published doesn't mean they're "imbecilities."

Do you prefer crankeries?

> This is clear from your claims that you have "pointed out" problems when, in fact, you
> did nothing of the sort.

Actually, I did. You are unable to construct a correct Minkowski diagram (fact pointed out not only by me but also by several others).
You are misinterpreting the Katrin results. Mas of the neutrino is not imaginary, neutrino does not travel at supraluminal speeds.
You are misinterpreting the mainstream papers that show that tachion communication violates causality.

This is why your garbage sits in the vixra cesspool. Where it belongs.

Gary Harnagel wrote:

>> > Now don't pull that baloney that the m² measurements are the
>> > difference between the neutrino flavors.
>>
>> They are.
>
> Wongo, DON'tknOw. YOU are the only fool promulgating this pernicious
> poop. Seems like YOU:

I remember it was deep *ILLEGAL* to promote drugs and medical advice,
mainstream media platforms, not being a doctor.

they're at war with you and want you dead
https://www.bitchute.com.video/hB03usomu8sY/

Boris Has A New Years Message For Us
https://www.bitchute.com/video/6lhLU-ledno/

Former Malaysia Prime Minister - Dr. Mahathir, talks about a one World
Government ruled by the wealth
https://www.bitchute.com/video/8n8LhD9VzHXf/

FTL has never existed as real in math.
That atom cannot be pushed to the speed of light.

Mitchell Raemsch

Richard Hachel wrote:

> Consider the following idea: there is no such thing as a natural number
> square that is double another natural number square.

This is easy to prove¹ (your "proofs" by example are not valid proofs),
but does not have anything to do with tachyons.

PointedEars
___________
¹ ∀ n, k ∈ ℕ, k > 0:
n² < 2 (n + k)² = 2 (n² + 2 n k + k²) = 2 n² + 4 n k + 2 k²

is true since already

n² < 2 n²
⇔ 1 < 2

is true. ∎
--
«Nec fasces, nec opes, sola artis sceptra perennant.»
(“Neither high office nor power, only the scepters of science survive.”)

—Tycho Brahe, astronomer (1546-1601): inscription at Hven

Why can't ;propulsion create a tachyon?
Atoms can be pushed by heat but not to the speed of light...
NO FTL...

Thomas 'PointedEars' Lahn wrote:

> Richard Hachel wrote:
>> Consider the following idea: there is no such thing as a natural number
>> square that is double another natural number square.
>
> This is easy to prove¹ (your "proofs" by example are not valid proofs),
> but does not have anything to do with tachyons.
> […]
> ___________
> ¹ ∀ n, k ∈ ℕ, k > 0:
> n² < 2 (n + k)² = 2 (n² + 2 n k + k²) = 2 n² + 4 n k + 2 k²
>
> is true since already
>
> n² < 2 n²
> ⇔ 1 < 2
>
> is true. ∎

To complete the proof, we also have to consider that the “(an)other natural
number” could be smaller than the “(a) natural number” (n). In that case we
have to prove that

∀ n, k ∈ ℕ, k > 0: n² ≠ 2 (n − k)².

Proof by contradiction: Assume that the opposite would be true, i.e. that

∃ k ∈ ℕ, k > 0: n² = 2 (n − k)²,

then

n² = 2 (n − k)²
= 2 (n² − 2 n k + k²)
= 2 n² − 4 n k + 2 k²
⇔ 0 = n² − 4 n k + 2 k²
= (n − 2 k)² − 2 k²
⇔ 2 k² = (n − 2 k)²
⇔ ±√2 k = ±(n − 2 k).

This is false because √2 ∉ ℕ (as √2 ∈ ℝ∖ℕ); k ∈ ℕ ⇒ √2 k ∉ ℕ. But
(n − 2 k) ∈ ℕ, so the LHS and the RHS cannot be equal (contradiction).
So they must be not equal. ∎

PointedEars
--
I heard that entropy isn't what it used to be.

(from: WolframAlpha)

On Saturday, January 1, 2022 at 7:19:19 PM UTC-8, Thomas 'PointedEars' Lahn wrote:
>> There is no such thing as a natural number square that is
>> double another natural number square.
>
> [proof: ] because √2 ∉ ℕ (as √2 ∈ ℝ∖ℕ); k ∈ ℕ ⇒ √2 k ∉ ℕ. But
> (n − 2 k) ∈ ℕ, so the LHS and the RHS cannot be equal (contradiction).

Well, it isn't necessary for √2 to be an integer in order for n√2 to be an integer, because n could be divisible by the denominator of √2. So what you really need to say is that √2 is not rational, meaning it is not the ratio of two integers, but that is simply re-stating the proposition, i.e., there are no integers m,n such that 2n^2 = m^2. The usual proof, going back to Euclid, essentially relies on unique factorization:

The power of 2 dividing m^2 is even, but the power of 2 dividing 2n^2 is odd, so m^2 cannot equal 2m^2.

On Saturday, January 1, 2022 at 7:19:19 PM UTC-8, Thomas 'PointedEars' Lahn wrote:
>> There is no such thing as a natural number square that is
>> double another natural number square.
>
> [proof: ] because √2 ∉ ℕ (as √2 ∈ ℝ∖ℕ); k ∈ ℕ ⇒ √2 k ∉ ℕ. But
> (n − 2 k) ∈ ℕ, so the LHS and the RHS cannot be equal (contradiction).

It isn't necessary for √2 to be an integer in order for n√2 to be an integer, because n could be divisible by the denominator of √2. So what you really need to say is that √2 is not *rational*, meaning it is not the ratio of two integers, but that is simply re-stating the proposition, i.e., there are no integers m,n such that 2n^2 = m^2. The usual proof, going back to Euclid, essentially relies on unique factorization:

The power of 2 dividing m^2 is even, but the power of 2 dividing 2n^2 is odd, so m^2 cannot equal 2n^2.

On Saturday, January 1, 2022 at 11:01:16 AM UTC-7, mitchr...@gmail.com wrote:
>
> FTL has never existed as real in math.

Math is not reality.

“as far as the laws of mathematics refer to reality, they are not certain; and as
far as they are certain, they do not refer to reality” -- Albert Einstein

> That atom cannot be pushed to the speed of light.
>
> Mitchell Raemsch

You completely misunderstand. No one is trying to push an atom "to the speed
of light.

Townes Olson wrote:

> On Saturday, January 1, 2022 at 7:19:19 PM UTC-8, Thomas 'PointedEars'
> Lahn wrote:
>>> There is no such thing as a natural number square that is
>>> double another natural number square.
>>
>> [proof: ] because √2 ∉ ℕ (as √2 ∈ ℝ∖ℕ); k ∈ ℕ ⇒ √2 k ∉ ℕ. But
>> (n − 2 k) ∈ ℕ, so the LHS and the RHS cannot be equal (contradiction).
>
> Well, it isn't necessary for √2 to be an integer in order for n√2 to be an
> integer, because n could be divisible by the denominator of √2.

No; you have overlooked that the stated premise of the proof is that n is a
natural number. Natural numbers are not divisible by √2 (which you probably
mean; ”divisible by the denominator of √2” does not make sense as √2 is not
a rational number – there is no fraction to express this number that could
have a denominator).

PointedEars
--
“Science is empirical: knowing the answer means nothing;
testing your knowledge means everything.”
—Dr. Lawrence M. Krauss, theoretical physicist,
in “A Universe from Nothing” (2009)

Townes Olson wrote:

> On Saturday, January 1, 2022 at 7:19:19 PM UTC-8, Thomas 'PointedEars'
> Lahn wrote:
>>> There is no such thing as a natural number square that is
>>> double another natural number square.
>>
>> [proof: ] because √2 ∉ ℕ (as √2 ∈ ℝ∖ℕ); k ∈ ℕ ⇒ √2 k ∉ ℕ. But
>> (n − 2 k) ∈ ℕ, so the LHS and the RHS cannot be equal (contradiction).
>
> It isn't necessary for √2 to be an integer in order for n√2 to be an
> integer, because n could be divisible by the denominator of √2. […]

AISB, n is a natural number.

Repeating nonsense does not make it true.

PointedEars
--
Heisenberg is out for a drive when he's stopped by a traffic cop.
The officer asks him "Do you know how fast you were going?"
Heisenberg replies "No, but I know where I am."
(from: WolframAlpha)

Thomas 'PointedEars' Lahn wrote:

> No; you have overlooked that the stated premise of the proof is that n is
> a natural number. Natural numbers are not divisible by √2 (which you
> probably mean; ”divisible by the denominator of √2” does not make sense as
> √2 is not a rational number – there is no fraction to express this number
> that could have a denominator).

I mean “no simple fraction”, of course.

PointedEars
--
“Science is empirical: knowing the answer means nothing;
testing your knowledge means everything.”
—Dr. Lawrence M. Krauss, theoretical physicist,
in “A Universe from Nothing” (2009)

On Sunday, January 2, 2022 at 2:52:32 PM UTC-8, Thomas 'PointedEars' Lahn wrote:
>>>> There is no such thing as a natural number square that is
>>>> double another natural number square.
>>>
>>> [proof: ] because √2 ∉ ℕ (as √2 ∈ ℝ∖ℕ); k ∈ ℕ ⇒ √2 k ∉ ℕ. But
>>> (n − 2 k) ∈ ℕ, so the LHS and the RHS cannot be equal (contradiction).
>>
>> It isn't necessary for √2 to be an integer in order for n√2 to be an
>> integer, because n could be divisible by the denominator of √2.
>
> You have overlooked that the stated premise of the proof is that n is a
> natural number.

No, your purported "proof" (shorn of superfluous fluff) was that the relation n√2 = m (where n and m are both natural numbers) is a contradiction because it would imply that √2 is a natural number. There are two things wrong with your reasoning, both explained in my post. First, the relation doesn't imply that √2 is a natural number, it implies that √2 is a rational number. Second, this is just re-stating the proposition to be proven, i.e., that there are no naturals such that 2n^2 = m^2, so your purported "proof" (after you correct your first trivial mistake) amounts to saying that √2 is irrational because √2 is irrational. In other words, your "reasoning" - even after fixing your mistake - is not a proof at all.

Lastly, I pointed out the usual trivial proof, going back to Euclid, based on what we call unique factorization, according to which the power of 2 dividing m^2 is even, but the power of 2 dividing 2n^2 is odd, so m^2 cannot equal 2n^2.

Now do you understand?