gen-define-diophantine-equations-and-give-one-short-worked-examp-b91c

Define Diophantine equations and give one short worked example.

# Diophantine Equations **Definition:** A Diophantine equation is a polynomial equation in one or more variables where we seek **integer solutions only** (sometimes restricted to positive integers). They're named after the ancient Greek mathematician Diophantus. The general form is: P(x₁, x₂, ..., xₙ) = 0, where P is a polynomial with integer coefficients and we want x₁, x₂, ..., xₙ ∈ ℤ. ## Worked Example **Solve:** 3x + 5y = 11 for integers x and y. **Solution:** Rearrange for x: $$x = \frac{11 - 5y}{3}$$ For x to be an integer, (11 - 5y) must be divisible by 3. Test values of y: - y = 1: x = (11 - 5)/3 = 6/3 = 2 ✓ - y = 2: x = (11 - 10)/3 = 1/3 ✗ - y = 4: x = (11 - 20)/3 = -9/3 = -3 ✓ **General solution:** Since gcd(3,5) = 1 divides 11, infinitely many solutions exist: - x = 2 - 5t - y = 1 + 3t where t ∈ ℤ. **Check (t=0):** 3(2) + 5(1) = 6 + 5 = 11 ✓