Solving Diophantine equations

### 6.1.23 Solving Diophantine equations

The isolve command attempts to solve the given equations over the integers. Note that it automatically solves for all of the indeterminates present in the equations.

- isolve takes one mandatory argument and two optional arguments:
  - eq, an equation or list of equations.
  - Optionally, symb, a (sequence or list of) symbol(s) which are used as the names for global variables present in the solution. These names default to _Z0,_Z1,… for general integers and to _N0,_N1,… for positive integers.
  - Optionally, seq=false, which makes isolve return only particular/fundamental solution(s) found by the solver. By default, seq=true, which makes isolve return sequences (classes) of solutions whenever possible.
- isolve can solve the following types of equations:
  - (systems of) linear equation(s).
  - general quadratic equations with two indeterminates.
  - equations of the type Q(x,y,z)=0, where Q is a ternary quadratic form.
  - equations of the type f(x)=g(y), where f,g∈ℤ[X] are monic polynomials with degrees m and n such that gcd(m,n)>1 and f(x)−g(y) is irreducible in ℚ[X,Y].

#### Examples

Linear equations and systems can be solved.

isolve(5x+42y+8=0)

| ⎡ ⎣ | x=−10+42_Z0,y=1−5_Z0 | ⎤ ⎦

isolve([x+y-z=4,x-2y+3z=3],m)

| ⎡ ⎣ | x=m,y=−4 m+15,z=−3 m+11 | ⎤ ⎦

To find the general solution to Pell-type equation x2−23y2=1:

sol:=isolve(x^2-23y^2=1,n)

| ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ | x= |
⎛ ⎜ ⎝ | 24+5 | √ |
23
| ⎞ ⎟ ⎠ |
n
| + | ⎛ ⎜ ⎝ | 24−5 | √ |
23
| ⎞ ⎟ ⎠ |
n
2
| ,y= |
⎛ ⎜ ⎝ | 24+5 | √ |
23
| ⎞ ⎟ ⎠ |
n
| − | ⎛ ⎜ ⎝ | 24−5 | √ |
23
| ⎞ ⎟ ⎠ |
n
2 | √ |
23
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

To check that it is indeed the solution, enter:

simplify(subs(x^2-23y^2-1,sol))

0 |  |  |  |  |  |  |  |  |  |

Now to obtain e.g. the first four solutions, enter:

simplify(apply(unapply(apply(rhs,sol),n),[1,2,3,4]))

| ⎡ ⎣ | [24,5],[1151,240],[55224,11515],[2649601,552480] | ⎤ ⎦

To obtain only the fundamental solution, enter:

isolve(x^2-23y^2=1,seq=false)

[x=24,y=5] |  |  |  |  |  |  |  |  |  |

The following two examples demonstrate solving quadratic equations with two indeterminates.

isolve(x^2-5x*y+4y^2=16)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
x=−4, | y=−5
x=0, | y=−2
x=4, | y=0
x=10, | y=2
x=21, | y=5
x=4, | y=5
x=0, | y=2
x=−4, | y=0
x=−10, | y=−2
x=−21, | y=−5
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

isolve(x^2-3x*y+y^2-x=2,n)

[[x |
= |
⎛ ⎜ ⎜ ⎝ |
√ |
5
| −3
2
| ⎞ ⎟ ⎟ ⎠ |
n
|  | ⎛ ⎜ ⎝ | −15 | √ |
5
| −33 | ⎞ ⎟ ⎠
10
| + |
⎛ ⎜ ⎜ ⎝ |
− | √ |
5
| −3
2
| ⎞ ⎟ ⎟ ⎠ |
n
|  | ⎛ ⎜ ⎝ | 15 | √ |
5
| −33 | ⎞ ⎟ ⎠
10
| − |
2
5
| ,
y |
= |
⎛ ⎜ ⎜ ⎝ |
√ |
5
| −3
2
| ⎞ ⎟ ⎟ ⎠ |
n
|  | ⎛ ⎜ ⎝ | −3 | √ |
5
| −6 | ⎞ ⎟ ⎠
5
| + |
⎛ ⎜ ⎜ ⎝ |
− | √ |
5
| −3
2
| ⎞ ⎟ ⎟ ⎠ |
n
|  | ⎛ ⎜ ⎝ | 3 | √ |
5
| −6 | ⎞ ⎟ ⎠
5
| − |
3
5
| ]]

isolve(8x^2-24x*y+18y^2+5x+7y+16=0)

⎡ ⎢ ⎣ |
x=−174 _Z12+17_Z1−2, | y=−116 _Z12+21_Z1−2
x=−174 _Z12+41_Z1−4, | y=−116 _Z12+37_Z1−4
| ⎤ ⎥ ⎦

Integral zeros of ternary quadratic forms can be found.

isolve(x^2+11y^2+6x*y-3z^2=0,a,b,c)

| ⎡ ⎣ | x=c | ⎛ ⎝ | −44 a2+12 b2 | ⎞ ⎠ | ,y=c | ⎛ ⎝ | 13 a2−3 b2−6 a b | ⎞ ⎠ | ,z=c | ⎛ ⎝ | −11 a2−3 b2+2 a b | ⎞ ⎠ | ⎤ ⎦

The components of the above solution can be divided by the GCD of −44 a2+12 b2, 13 a2−3 b2−6 a b, and −11 a2−3 b2+2 a b, thus producing a parametrization for the pairwise-coprime solutions given c=1.

Certain polynomial equations of the type f(x)=g(y) can be fully solved, as in the following example.

isolve(x^2-3x+5=y^8-y^7+9y^6-7y^5+4y^4-y^3)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
x=−3, | y=−1
x=6, | y=−1
x=0, | y=1
x=3, | y=1
x=660, | y=5
x=−657, | y=5
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

The above list contains all integer solutions to the given equation.