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How do you solve something like: 3a + 4.2b + 7c + 10.5d = 59.3 over the integers?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

Hi;

What is it for because that will tell me what type of solutions are allowed?

Here is a solution

a = 6, b = 4, c = 2, d = 1

If we allow negative numbers ( they are integers ) then there is likely an infinite number of solutions.

The problem is much more interesting when defined as 0 ≤ a,b,c,d Then it is easy to prove that is the only solution.

**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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Yes! They wanted positive int solutions.

Did you do it with PSLQ?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

Yes a PSLQ can do that and is the method of choice. Then you use generating functions to prove it is the only solution.

So you see when you said "mere computation" in the other thread I was glad when you posted this. Numerics and computation are not mere but actually more powerful than classical mathematics. [comment removed]

This is only one example of its power. Diophantine equations, are the secret to doing powerful combinatorics.

**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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I do not know if it is classical mathematics or not but let us keep the debate for later.

How to do this by hand? Please teach

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

Hmmm, how do you define by hand? Where is the problem from?

Here is a good rule:

school problems = hand methods

Everything else in the universe can not be done by hand.

**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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brilliant.

Please tell me how you would do it if you lived in the nineteenth century.

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

So, you have put me back into the 19th century. First of all, I want to tell you I do not like that century. Have I ever told you the story of how grandpappym killed 30 Union soldiers with just his revolver a bayonet and his hands? It was September of 1863 ( the nineteenth century ) at the battle of Chickamagua, but wait I am getting off the point.

First you could bring it into the integers by multiplying by 10.

30a + 42b + 70c + 105d = 593

Then I would proceed empirically from the b. It is obvious that b can only be 4 or 9. You are done right from there! Do you see why?

**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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Empirically? ??

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

The other three numbers are all divisible by 5. So some multiple of 593 - 42 b must also be divisible by 5. Only 2 numbers 4 and 9 for b do that.

**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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Why not anything greater than that? and only that?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

{551,509,467,425,383,341,299,257,215,173,131,89,47,5,-37}

That is the result of 593 - 42b for b = 1 to 15. It is only arithmetic and can be done in your head in less than 5 minutes. You see that only

593 - 4 x 42 = 425

593 - 9 x 42 = 215

Those are the only 2 divisible by 5.

Now you cannot partition 215 using { 30, 70, 105 } so 4 must be the answer.

**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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hmm...

next,please;

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

Easy to see that 215 can not be the sum of 30,70 and 105

So you have a new equation:

30 a + 70 c + 105 d = 425

divide by 5:

6a + 14c + 21d = 85

Again 6 and 21 are divisible by 3 and 14 isn't so pick out the14:

85 - 14 c = {71,57,43,29,15,1} for c = 1,2,3,4,5,6

So only c =2 and c = 5 yields a number that is divisible by 3. Picking the 5 for c we

6a + 21 d = 15 which is impossible so c = 2 and you are left with this diophantine equation:

6 a + 21 d = 57

That is easy to do in your head. If you can not then continue as I have done,

a = 6, d = 1, c = 2, b =4

**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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That is easy to do in your head.

I could do that, but the rest were tough.

What is the computer method for this?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

What part?

Computer methods are basically the same as shown but thousands of times faster and with no errors.

**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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How do you write a PSLQ?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

I think anonimnystefy and I were discussing that same problem. I do not even remember what the answer was.

**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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But you certainly remember the algorithm

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

The algorithm is written in M. It uses a built in command called Lattice Reduce. That is what I can not do.

There is a whole thread on it.

**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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Why did you link me to page 11?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

It is a whole discussion of Lattice Reduction.

You said some part of the above solution was tough. Which one? I will explain it further.

**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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I can understand it but I don't think I will be able to do the same for any other problem

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,243

That method like all book methods is only possible for a few types of problems. I could easily change the problem to be intractable.

The first rule of a place like brilliant or any textbook for that matter is that the problem given must have a solution. Knowing that, you can look for one.

**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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