Diophantine Equation #diophantine #equation
Diophantine Equation : A Diophantine equation is a polynomial equation in one or more variables where only integer solutions are sought . General form: f ( x 1 , x 2 , … , x n ) = 0 f(x_1, x_2, \dots, x_n) = 0 f ( x 1 , x 2 , … , x n ) = 0 where: f f f has integer coefficients Solutions must be integers (sometimes non-negative integers) 📌 Named after Diophantus of Alexandria (3rd century). 2. Types of Diophantine Equations 2.1 Linear Diophantine Equations Form: a x + b y = c ax + by = c a x + b y = c where a , b , c ∈ Z a, b, c \in \mathbb{Z} a , b , c ∈ Z Condition for solutions: gcd ( a , b ) ∣ c \gcd(a, b) \mid c g cd ( a , b ) ∣ c General solution: If d = gcd ( a , b ) d = \gcd(a, b) d = g cd ( a , b ) , and ( x 0 , y 0 ) (x_0, y_0) ( x 0 , y 0 ) is one solution, then: x = x 0 + b d t , y = y 0 − a d t , t ∈ Z x = x_0 + \frac{b}{d}t,\quad y = y_0 - \frac{a}{d}t,\quad t \in \mathbb{Z} x = x 0 ...