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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,758

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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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: 90,758

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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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: 90,758

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,758

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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