Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Number-theory problem (from another forum)

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

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."

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

Okay.

---

"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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

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.

---

"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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

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)

---

"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."

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

Hello everybody,

Any updates on this?

---

"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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

Hi bobbym,

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

---

"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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

Yes, that's really odd.
And 40 doesn't even seem to be related to 17.

---

"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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

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

---

"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."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number-theory problem (from another forum)

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.

Avon

Member

Registered: 2007-06-28

Posts: 80

Re: Number-theory problem (from another forum)

The first two congruences are equivalent to

.

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

and

.

Hence if

then

and if

then

.

Now suppose that |m| > 17 and |n| > 17.
It follows from the two congruences above that

so

.

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Number-theory problem (from another forum)

Hi Avon,

Thanks!

---

"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."