Linear Diophantine Equations

A linear equation ax + by = c, with a ≠ o, b ≠ o and c integers is called a linear Diophantine equation in two unknown x and y.

Solution of linear Diophantine equation: A pair of integers x₀, y₀ is called a solution of ax + by = c if ax₀ + by₀ = c.

1. Let a ≠ o, b ≠ o and c be any three integers and d = (a, b). The linear Diophantine equation ax + by = c has a solution iff d/c.
2. If x₀, y₀ is any particular solution of ax + by = c then any other solution of this equation is x' = x₀ - (b/d)t, y' = y₀ + (a/d)t, t being any integer.
3. The Diophantine equation y² = x³ + k has no solution if k has the form k = (4n - 1)³ - 4m², where m and n are integers such that no prime p ≡ -1 (mod 4) divides m.