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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,656

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,656

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

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 isnt.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,656

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

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.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,656

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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