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August 2nd, 2010, 09:56 AM   #1
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diophantine equation

How to identify diophantine equations and how is it solved?

Please solve along with an example.
Thank you.
bentick is offline  
 
August 2nd, 2010, 12:31 PM   #2
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Re: diophantine equation

http://en.wikipedia.org/wiki/Diophantine_equation

Look at the above.
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August 2nd, 2010, 12:44 PM   #3
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Re: diophantine equation

A diophantine equation is one where the only allowed solutions are integers. For example:

In a pet shop, rats cost 5 dollars, guppies cost 3 dollars and crickets cost 10 cents.
One hundred animals are sold, and the total receipts are 100 dollars. How many rats,
guppies and crickets were sold?

You get the equations:
r + g + c = 100
5r + 3g + 0.1c = 100

This simplifies to the diophantine equation:
49r + 29g = 900

This is one equation in two unknowns and it normally has an infinite number of solutions. However, if you further restrict the problem so that c, r and g must be positive integers, then the solution can be found as follows:

The number of crickets sold must be a multiple of 10. Try the different values of c: 0, 10, 20, 30, ....., 100.
If c = 10, then r + g = 90. Try different values of r. For r=1, r=2, and so on, there are no integer solutions when c=10.

Continue trying c=20, 30, .....

When you get to c=80, then r + g = 20, and you eventually find that c=80, r=16, and g=4 is a solution where c, r and g are integers.

Really smart people are working on the theory of diophantine equations, but I think it is safe to say that finding solutions to a diophantine equation problem can be described as solving a puzzle.
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August 3rd, 2010, 01:41 AM   #4
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Re: diophantine equation

Can't such an equations,

be solved by:

...

...
Backsubstitute:

and


...


This is bizarre, I get the same, as well by backsubstituting the last fraction. Can anyone show me the error, or is the method wrong?

Hoempa
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August 3rd, 2010, 03:54 AM   #5
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It's not bizarre.

r + g + c = 100
50r + 30g + c = 1000

Subtracting the first equation from the second gives 49r + 29g = 900, so g = (900 - 49r)/29.

You correctly obtained the same result by a longer route.
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August 3rd, 2010, 09:05 AM   #6
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Re: diophantine equation

And so did followingmycoffeecup, with that method..
Thanks, skipjack

Hoempa
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August 4th, 2010, 05:17 AM   #7
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Re: diophantine equation

Thnx for the example.

But I still didn't get how to IDENTIFY a diophantine equation.
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August 5th, 2010, 08:21 AM   #8
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Re: diophantine equation

A diophantine equation is an equation where the roots must be integers. It depends on the problem that you are solving. In the example above the equations are:
r + g + c = 100
50r + 30g + c = 1000

You have two equations in three unknowns and there are an infinite number of solutions. If you restrict r, g, c to being positive integers, then it is diophantine. You can't buy half a rat. If it was a butcher shop instead of a pet store, then it would not be diophantine.
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