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

The following problem was posted by someone else on another forum but has not received any reply on that forum. Im re-posting it here so people here can have a crack at it.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hello JaneFairfax;

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hi all,

I don't know whether there are infinite integers satisfying the congruences.

All the 40 integers are in between -305 and 271.

I combined those to:

I'm stuck here!

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hi,

An update:

There were 4 duplicates in the answer I calculated, so there are 36 solutions.

All the answers are of the form:

I still can't prove that there are more solutions or not.

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi all;

A little bit more:

The system of quadratic congruences can be reduced down to a system of linear congruences.

As a consequence of the above congruences we can say.

And:

We can now form the 2 linear congruences:

With:

Which just means nm = {...-33, -16, 1,18,35,52,69,86,...}

Unfortunately, I still am unable to prove using this easier set of congruences that there are just 40 solutions.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hi bobbym,

From my previous post, is it possible to determine or prove that there are finite values of n satisfying the condition?

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi gAr;

I did not see that. I will look at it now.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Okay.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi gAr;

So far I am unable to go the last step. I am checking on CRT right now. One question before we move on.

Here is my 40, in (m,n) form. How are you getting 36?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hi bobbym,

Yes, those are the values I'm also getting. I made a mistake when comparing the methods.

I took n from the brute force method and put it in the fraction, and got 4 duplicate m's.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi gAr;

Your parametric form is beautiful but it does leave out some solutions.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Yes, sad!

But it yields only integers when the n from the answer is substituted, so that form is okay, I guess.

Need to patch it!

*edit: The 4 missing values from the parametric form:

(-18,1)

(-16,1)

(-1,16)

(1,18)

*Last edited by gAr (2011-06-10 00:39:31)*

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hello everybody,

Any updates on this?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi gAr;

Nothing on this end.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hi bobbym,

Okay, I wonder whether JaneFairfax or the other forum has got something.

Just curious.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi gAr;

I could not find the other forum. I tend to think the problem may have a mistake. Usually when someone posts a number theory question it has no solutions, a couple of solutions or an infinite number of solutions. It looks like this one has just 40 solutions, that is odd.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Yes, that's really odd.

And 40 doesn't even seem to be related to 17.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Also the fact that the 40 solutions occur so early and in such a small interval.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Yes!

I'll wait for JaneFairfax, she knows number theory more than I do.

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

That sounds like a good idea right now.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**Avon****Member**- Registered: 2007-06-28
- Posts: 80

The first two congruences are equivalent to

.

17n-1 and 17m+1 cannot be 0 so we have

.

Hence if

thenand if then .

Now suppose that |m| > 17 and |n| > 17.

It follows from the two congruences above that

.

17n-17m-1 cannot be 0 so we have

so

and hence .

Since |m|-17 and |n|-17 are both positive integers it follows that

and similarly .

Therefore if (m,n) is a solution to these congruences then |m| and |n| are both at most 307

and so the bobbym's brute force search has found all of the solutions.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,482

Hi Avon,

Thanks!

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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