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### Diophantine equations in Number Theory

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Date: 01/24/2001 at 14:51:39
From: Kristi Harp
Subject: Diophantine equations in number theory

Here is the problem:

If a and b are relatively prime positive integers, prove that the
Diophantine equation ax-by = c has infinitely many solutions in the
positive integers.

[Hint: There exist integers x0 and y0 such that ax0+by0 = c. For any
integer t, which is larger that both |x0|/b and |y0|/a, a positive
solution of the given equation is x = x0+bt, y = -(y0-at).]

(Note: x0 is x null or naught...I didn't know how else to write it.)

would greatly appreciate it. Thank you.
```

```
Date: 01/24/2001 at 19:35:01
From: Doctor Anthony
Subject: Re: Diophantine equations in number theory

If a and b are coprime you can always find integers p, q such that

ap + bq = 1  using Euclid's algorithm.

Then multiply by c to get

ax' + by' = c ......(1)  where  x' = cp,  y' = cq

and finally let x = x'+bt   y = -(y'-at)  and substituting into the
equation

ax - by = c  we get

a(x'+bt) +b(y'-at) = c

ax' + abt + by' - abt = c

ax' + by' = c    which we have shown from (1) is true.

It follows that there are an infinity of solutions of the form

x = x'+bt     y = -(y'-at)   where  ax'+by'=c

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations
High School Number Theory

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