Number Theory

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• state and prove Fermat's Little Theorem & its generalisation using Euler's function & use them to implement the RSA cipher & dicrete log cipher
• apply different methods of proof to verify mathematical assertions, including proof by induction, by contrapositive and by contradiction
• understand the basics of modular arithmetic
• solve systems of Diophantine equations using the Chinese Remainder Theorem & the Euclidean algorithm
• analyse hypotheses and conclusions of mathematical statements

• Proof and Mathematical Logic
• Number Theory
• Divisibility, least common multiples, Euclid's algorithm.
• Integer solutions of ax + by = c.
• Prime numbers and prime-power factorisations, irrational numbers.
• Existence of infinitely many primes.
• Modular arithmetic, linear congruences. Chinese Remainder Theorem.
• Fermat's Little Theorem.
• Units, Euler's function, Euler’s Theorem.
• Cryptography
• Diffie-Hellman-Merkle and Rivest-Shamir-Adleman key exchange systems
• Pythagorean triples & Fermat’s Last Theorem.

Lectures, problem classes, workshops, private study

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.