Definition:Diophantine Equation/Linear Diophantine Equation

Definition

A linear Diophantine equation is a Diophantine equation in which all the arguments appear to no higher than the first degree.

For example:

$ax + by + c = 0$

$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$

Examples

Example: $15 x + 27 y = 1$

The linear diophantine equation:

$15 x + 27 y = 1$

has no solutions for $x$ and $y$ integers.

Example: $5 x + 6 y = 1$

The linear diophantine equation:

$5 x + 6 y = 1$

has the general solution:

$x = -1 + 6 t, y = 1 - 5 t$

Also see

• Results about linear diophantine equations can be found here.

Source of Name

This entry was named for Diophantus of Alexandria.

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-3}$ The Linear Diophantine Equation