Diophantine Equation Examples

Number Theory · Axiom Academy

EXAMPLE Diophantine Equation Examples Master linear Diophantine equations through five worked examples Excellent work! You've mastered solving linear Diophantine equations. Here's what we learned: Existence Test: A linear Diophantine equation ax + by = c has integer solutions if and only if gcd(a, b) divides c . Finding Solutions: Use the Extended Euclidean Algorithm to find a particular solution, then scale it to match the constant term. General Solution: Once you have a particular solution (x₀, y₀) , all solutions are given by x = x₀ + (b/d)t and y = y₀ - (a/d)t where d = gcd(a, b) and t is any integer. No Solutions Case: If gcd(a, b) does not divide c , the equation has no integer solutions. Positive Solutions: To find positive solutions only, solve the inequalities x₀ + (b/d)t > 0 and y₀ - (a/d)t > 0 to determine valid values of t . Real-World Applications: Diophantine equations model coin problems, scheduling, and resource allocation where only whole numbers make sense. This systematic approach works for any linear Diophantine equation. Practice identifying whether solutions exist, finding particular solutions, and writing general solutions!

This is the written version of the interactive lesson above. See the full Number Theory course.