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Diophantine Equations in Three Variables

Date: 11/17/2004 at 08:19:26
From: Kazim
Subject: Diophantine Equations

I need to know how to get positive integer solutions of two 
Diophantine equations having three variables.  For example:

  2x + 3y + 7z = 32 ; 3x + 4y - z = 19 

(give the positive set of triples for the above equations)



Date: 11/17/2004 at 19:34:32
From: Doctor Vogler
Subject: Re: Diophantine Equations

Hi Kazim,

First of all, there are two ways to interpret your two equations, and
I'm not sure which one you want.  Either you want positive integer
triples (x, y, z) that satisfy *both* equations

  2x + 3y + 7z = 32
  3x + 4y - z = 19,

or you want all positive integer triples that satisfy the equation

  2x + 3y + 7z = 32

as well as all positive integer triples that satisfy the equation

  3x + 4y - z = 19.

If the first case is what you want, then you just solve for the
variable z in the second equation, substitute into the first equation,
and that gives you a normal linear Diophantine equation in two 
variables.  (If there weren't a variable with a coefficient of 1, then
you could make one of the coefficients 1--leaving the rest integers--
by adding and subtracting multiples of one equation from the other,
back and forth.)

Here's an answer for our archives that discusses this case:

  Systems with More Variables than Equations
    http://mathforum.org/library/drmath/view/61825.html 

If the second case is what you want, then you are asking how to solve
a linear Diophantine equation in three variables.  The following
answer discusses this case:

  Diophantine Equations, Step by Step
    http://mathforum.org/library/drmath/view/61325.html 

However, I would like to take this opportunity to describe a different
technique for solving three-variable linear Diophantine equations that
gives an answer more like the formula for all integer solutions to
two-variable linear Diophantine equations.

So I will use this technique to find all integer solutions to an
equation, and then mention how you use this to find all *positive*
integer solutions.

First let's solve the equation

  6x + 10y + 15z = 0.

We know that z has to be even, right?  And we know that y has to a
multiple of 3, right?  So suppose that

  z = 2a

and

  y = 3b.

Then we solve for x:

  6x + 10(3b) + 15(2a) = 0

  6x = -30b - 30a

  x = -5(a+b).

Then that means that all solutions are

  x = -5(a+b)
  y = 3b
  z = 2a

for any integers a and b.

You can describe all of the solutions in different ways, as well. 
Basically, x has to be divisible by 5, y by 3, and x by 2.  Then you
can pick any two of those variables to be anything (satisfying the
divisibility condition) and the third variable is determined from the
equation.

Let's try another.  Let's solve your equation

  2x + 3y + 7z = 32.

As in solving a two-variable linear Diophantine equation, the first
thing to do is find any *one* integer solution.  In three variables,
the easiest way to do this is often to pick any number for one of the
variables and then use your usual techniques (such as modular inverses
or the Euclidean algorithm) to find solutions in the others.  Here's
an easy solution for this equation:  Take x=16 to cancel the 32 on the
right, so you have

  3y + 7z = 0,

and then take y = 7 and z = -3.  So now we have one solution,

  (16, 7, -3)

and we want to find all solutions.  Well, let's suppose that

  (x, y, z)

is another solution.  Then what happens if we subtract our known
solution from it?

  2(x - 16) + 3(y - 7) + 7(z + 3) = (2x + 3y + 7z) - 32 = 0.

So that means that if

  a = x - 16
  b = y - 7
  c = z + 3,

then (a, b, c) satisfy the equation

  2a + 3b + 7c = 0.

Now you see why I started with an equation that equaled zero.  In the
previous case, there were common factors between any two of the
coefficients, and that allowed us to divide by those common factors. 
If there are not, as here, then you don't divide by anything.  (I.e.
you divide by the common factor 1.)

But we still can't solve for a until we know that

  3b + 7c

is even.  So we have to make it even.  We do this by noticing that if

  3b + 7c

is even, then so is b - c.  So we let

  b = c + 2r,

and now we know that

  2a = -(3b + 7c) = -(3c + 6r + 7c) = -(10c + 6r)

is even, so we can solve for a:

  a = -(5c + 3r).

And c can be anything.  So we have

  a = -5s - 3r
  b = s + 2r
  c = s

for some integers r and s.  That is, every solution to

  2a + 3b + 7c = 0

has that form for some r and s.  Furthermore, for any integers r and
s, those three numbers are solutions.  If we recall that

  a = x - 16
  b = y - 7
  c = z + 3,

then we can also say about all solutions to

  2x + 3y + 7z = 32

that they are given by

  x = 16 - 5s - 3r
  y = 7 + s + 2r
  z = -3 + s.

Finally, if we want to find only positive solutions, then we just have
to find all (s, r) pairs that satisfy the three inequalities

  x = 16 - 5s - 3r > 0
  y = 7 + s + 2r > 0
  z = -3 + s > 0.

The easiest way to solve these inequalities might be to graph them (as
lines) on a plane and count all of the lattice points (integer points).

Now, let me summarize (and generalize).  Suppose we want to solve an
equation of the form

  Ax + By + Cz = N

for some (known) integers A, B, C, and N.  We may assume that A, B,
and C have no common factor (of all three), since if such a common
factor divides N, then we can divide the whole equation by that
number.  If that common factor does not divide N, then there are no
solutions.  First we find a single particular solution by choosing any
number x.  If gcd(B, C) > 1, then our x must satisfy

  Ax = N (mod gcd(B, C)).

Then we find some solution for y and z, such as by using the Euclidean
Algorithm.  So now we have a particular solution (x0, y0, z0).  To
find all solutions, we let

  x' = x - x0
  y' = y - y0
  z' = z - z0

and find that (x', y', z') must satisfy

  Ax' + By' + Cz' = 0.

Next, x' must be divisible by gcd(B, C), and y' must be divisible by
gcd(A, C), and z' must be divisible by gcd(A, B), so we let

  x" = x'/gcd(B, C)
  y" = x'/gcd(A, C)
  z" = z'/gcd(A, B),

and we let

  A' = A/(gcd(A, C)*gcd(A, B))
  B' = B/(gcd(A, B)*gcd(B, C))
  C' = C/(gcd(A, C)*gcd(B, C))

so that now we have

  A'x" + B'y" + C'z" = 0

where A', B', and C' are pairwise relatively prime (that is, no two
have a common factor).

Now we would like to solve for x", so we decide what b must be mod A'.

  B'y" + C'z" = 0 (mod A').

So we find an integer k that satisfies

  B'k + C' = 0 (mod A'),

which is -C'/B' (mod A') and can be computed using the Euclidean
algorithm, and then y" - kz" must be divisible by A', so we let

  y" = kz" + A'r

and then we solve for x"

  A'x" = -B'y" - C'z"
       = -B'(kz" + A'r) - C'z"
       = -(B'k + C')z" - A'B'r

and so

  x" = -B'r - z"(B'k + C')/A'

where that last number m = (B'k + C')/A' is an integer.  Finally, we
choose any number for z" and we get

  x" = -B'r - ms
  y" = ks + A'r
  z" = s

  x' = gcd(B, C) * (-B'r - ms)
  y' = gcd(A, C) * (ks + A'r)
  z' = gcd(A, B) * s

  x = x0 + gcd(B, C) * (-B'r - ms)
  y = y0 + gcd(A, C) * (ks + A'r)
  z = z0 + gcd(A, B) * s

and this last formula gives us all integer solutions to the equation

  Ax + By + Cz = N

in the form of linear combinations of the two arbitrary integers r and
s.  Of course, this isn't the *only* way to express all integer
solutions.  For example, we can change r to -t or t-s or t-7s, and get
new ways to write out all integer solutions.  Or we can change s to
t-6r, for example.  (We can't change them to just anything, but we can
change them to many different things.)

I hope you find this useful and informative.  If you have any
questions about this or need more help, please write back, and I will
try to explain further.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Number Theory
High School Number Theory

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