# Theory Of Numbers –

### mth415

(3 credits)

This course is an introduction to the main concepts of number theory. The topics will include divisibility of numbers, prime numbers, Euclid’s theorem and algorithm, fundamental theory of arithmetic, the sequence of primes, linear congruence, solving polynomials congruence, Fermat’s theorem, quadratic residuals, and roots of congruences. Students will deepen their experience with axiomatic systems.
This undergraduate-level course is 5 weeks. This course is available to take individually or as part of a degree or certificate program. To enroll, speak with an Enrollment Representative.

### Integers

• Verify properties of numbers and sequences.
• Find sums and products of numbers.
• Use the principles of mathematical induction to complete positive integer exercises.
• Implement the Fibonacci sequence.
• Employ the concept of divisibility of one integer by another integer.
• Perform arithmetic operations with integers using algorithms.

### Primes and Greatest Common Divisors

• Find the greatest common divisor of two integers.
• Use the Euclidean algorithm to produce the greatest common divisor.
• Apply the fundamental theorem of arithmetic to find prime factorization of positive integers.
• Perform factorization of Fermat numbers.
• Solve linear diophantine equations.
• Convert regular and decimal fractions into each other.

### Congruences

• Employ modular arithmetic to complete problem sets.
• Solve linear congruences.
• Apply the Chinese remainder theorem to solve systems of linear congruences.
• Solve polynomial congruences.

### Special Congruences and Nonlinear Diophantine Equations

• Demonstrate the validity of Wilson’s theorem.
• Use Fermat’s little theorem to find the least positive residue.
• Evaluate the Euler phi-function and solve equations involving the function.
• Determine the sum and number of divisors of positive integers.
• Find the Pythagorean triples.
• Solve some nonlinear diophantine equations as special cases of Fermat's last theorem.

### Cryptology

• Perform encryption and decryption on cipher text.
• Apply the variety of ciphers to encrypt and decrypt messages.
• Use the RSA cryptosystem to encrypt and decrypt messages.
• Implement cryptosystems to cryptographic applications.