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#1 2009-12-27 05:50:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Fumble and Botch strike again.

Hi;

It is well known that:

Prove that:

Has no solution in integers:

Proof:

8x^2 is congruent to {0,2,5,8} mod 9

270y^4 is congruent to 0 mod 9

So 8x^2 + 270y^4 is congruent to {0,2,5,8} mod 9

While 210 is congruent to 3 mod 9:

So 8x^2 + 270y^4 = 210 has no solutions in integers.

Total garbage! What is wrong with this?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#2 2009-12-27 07:15:15

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Fumble and Botch strike again.

As far as I can tell the proof is sound. The statement is certainly true, and here's a simpler way of showing it:

If y is a non-zero integer, then y^4 is a positive integer.
Also, x^2 is non-negative for all integers x, and so we must have 270y^4 = 0 if there is any chance of solving the equation.

However, we are then left with x^2 = 210/8, which has no integer solutions.


Why did the vector cross the road?
It wanted to be normal.

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#3 2009-12-27 07:23:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: Fumble and Botch strike again.

Hi;

Confucius wrote:

Do not use a cannon to kill a mosquito

That is exactly what is wrong with it. Lot of typing no thinking.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#4 2009-12-27 09:26:34

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Fumble and Botch strike again.

and
and so for the equation to have any solution, we must have
. That means
, which means
and so
. Then the LHS would be divisible by 9 whereas the RHS isn’t.

Last edited by JaneFairfax (2009-12-27 09:26:54)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#5 2009-12-27 12:52:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: Fumble and Botch strike again.

Hi Jane;

Yes, the poster missed that too. He seems to love mods, uses them to prove or disprove everything.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#6 2009-12-28 04:56:35

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Fumble and Botch strike again.

There is nothing wrong with mods; I use them a lot too. But I also like to keep my mind open for simple solutions, if they exist. smile


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#7 2009-12-28 18:49:13

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: Fumble and Botch strike again.

Hi Jane;

If the simple solution is the only one available, then I am sure you will use it. No insult intended but my assessment of you is that your artistic side, dominates.

In case I don't see you, have a happy New Year!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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