Diophantine Equation Solver
Diophantine Equation Solver finds integer solutions to linear equations of the form \( ax + by = c \) with solution plotting.
Diophantine Equation Solver
Finds integer solutions to the linear Diophantine equation:
Equation: \( ax + by = c \)
Conditions for Solvability:
- Solutions exist if and only if \(\gcd(a, b)\) divides \( c \).
- If \(\gcd(a, b) = d\) and \( d \mid c \), a particular solution can be found.
- General solution: If \( (x_0, y_0) \) is a particular solution, then: \[ x = x_0 + \frac{b}{d} k, \quad y = y_0 – \frac{a}{d} k, \quad k \in \mathbb{Z} \]
Where:
- \( a, b \): Integer coefficients
- \( x, y \): Integer variables
- \( c \): Integer constant
- \( \gcd(a, b) \): Greatest common divisor
- \( k \): Integer parameter
Diophantine Equation Solver
Diophantine Equation Solver (Example 1)
Equation: \( 3x + 6y = 9 \)
\[
\gcd(3, 6) = 3, \quad 3 \mid 9
\]
\[
x_0 = 3, y_0 = 0 \quad (\text{particular solution})
\]
\[
x = 3 + 2k, \quad y = -k, \quad k \in \mathbb{Z}
\]
Diophantine Equation Solver (Example 2)
Equation: \( 4x + 6y = 10 \)
\[
\gcd(4, 6) = 2, \quad 2 \mid 10
\]
\[
x_0 = 2, y_0 = 1 \quad (\text{particular solution})
\]
\[
x = 2 + 3k, \quad y = 1 – 2k, \quad k \in \mathbb{Z}
\]
Diophantine Equation Solver (Example 3)
Equation: \( 5x + 7y = 13 \)
\[
\gcd(5, 7) = 1, \quad 1 \mid 13
\]
\[
x_0 = 6, y_0 = -1 \quad (\text{particular solution})
\]
\[
x = 6 + 7k, \quad y = -1 – 5k, \quad k \in \mathbb{Z}
\]
Diophantine Equation Solver (Example 4)
Equation: \( 2x + 4y = 6 \)
\[
\gcd(2, 4) = 2, \quad 2 \mid 6
\]
\[
x_0 = 3, y_0 = 0 \quad (\text{particular solution})
\]
\[
x = 3 + 2k, \quad y = -k, \quad k \in \mathbb{Z}
\]