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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

I need to make a program that solves different diophantine equations.

Can someone help me?

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

1. ax+by=c

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

```
ax + by = c
by = -ax + c
y = (-a/b)x + (c/b)
So y = at + ?1? and
x = bt + ?2?
-------
Put into original equation ax + by = c
a(bt + ?2?) + b(at + ?1?) = c
abt + a?2? + bat + b?1? = c
2bat + a?2? + b?1? = c
---------
y=7x + .3
y=7t + ?
x=t + ?
No integer solutions for this 7x + .3
---------
No luck yet.
```

**igloo** **myrtilles** **fourmis**

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

For lineal d.e. we can use the this

*Last edited by krassi_holmz (2005-12-31 08:10:13)*

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

At wolfram's site, equation # 6 has a "+ 1" in it (Euclidian thing).

I wonder why it always works out to a one, like the example they have?

*Last edited by John E. Franklin (2005-12-31 06:33:03)*

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Also equation # 10 at their site has an incorrect sign, I think.

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

I found this a very useful page on this subject.

http://uzweb.uz.ac.zw/science/maths/zimaths/62/dioph.htm

or

My favorite part is at the end of the page:

the web page wrote:

(i) The equation 3x + 6y = 22 has no solution since (3, 6) = 3 does not divide 22.

(ii) The equation 7x + 11y = 13 has solution x = -39, y = 26. For

11 = 1·7 + 4, 7 = 1·4 + 3, 4 = 1·3 + 1.

Thus (7, 11) = 1, which divides 13. Further, working from the last equation back to the first,1 = 4 - 3 = 4 -(7-4) = 2·4 - 7 = 2·11 - 3·7.

Hence7·(-3) + 11(2) = 1, 7·(-39) + 11(26) = 13.

The other solutions are given byx = -39 + 11r, y = 26 - 7r

where r is any integer.

*Last edited by John E. Franklin (2005-12-31 09:32:35)*

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

OK. We have simple alogritm for ax+by. I know it as Euler's reduction algoritm.

The next general case:

ax^2+by=c

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