Solving Diophantine Equations: A Challenge Quiz

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1. Which ancient mathematician, author of the Arithmetica, gave his name to Diophantine equations?

   • A. Euclid
   • B. Diophantus
   • C. Archimedes
   • D. Pythagoras

2. What distinguishes a Diophantine equation from an ordinary algebraic equation?

   • A. Only rational (fractional) solutions are accepted
   • B. Only real solutions are accepted
   • C. Only integer solutions are accepted
   • D. Only complex solutions are accepted

3. What is the general form of a linear Diophantine equation in two unknowns?

   • A. ax^2 + by^2 = c
   • B. ax + by = c
   • C. ax^3 + by^3 = c
   • D. ax^4 + by^4 = c

4. Which result gives the condition for the linear Diophantine equation ax + by = c to have integer solutions?

   • A. Bezout's Identity
   • B. Fermat's Last Theorem
   • C. Euclid's Lemma
   • D. Gauss's Lemma

5. How does the Euclidean algorithm compute the greatest common divisor of two integers?

   • A. Prime factorization of both numbers
   • B. Synthetic division
   • C. Reduction modulo a fixed prime
   • D. Repeated division with remainders

6. Which theorem states that every positive integer can be written as a sum of four squares?

   • A. Lagrange's Four-Square Theorem
   • B. Gauss's Eureka Theorem
   • C. Fermat's Two-Square Theorem
   • D. Euler's Totient Theorem

7. Which of the following is a nonlinear Diophantine equation?

   • A. x + y = 10
   • B. x^2 + y^2 = 100
   • C. 3x + 4y = 12
   • D. 5x - 7y = 1

8. How is modular arithmetic most often used when analysing a Diophantine equation?

   • A. To prove no integer solutions exist by deriving a contradiction modulo some n
   • B. To compute the greatest common divisor of the coefficients
   • C. To convert integer solutions into rational ones
   • D. To lower the degree of the equation by one

9. Which pair is a solution to the Diophantine equation 2x + 3y = 7?

   • A. x = 2, y = 1
   • B. x = 1, y = 2
   • C. x = 3, y = 1
   • D. x = 1, y = 1

10. Which famous Diophantine problem was finally proved by Andrew Wiles in 1994?

    • A. Fermat's Last Theorem
    • B. Bezout's Identity
    • C. Gauss's Lemma
    • D. Euler's Conjecture on sums of powers

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