http://www.phoenix.edu
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.
To enroll, speak with an Enrollment Advisor.
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Primes and Greatest Common Divisors
- Convert regular and decimal fractions into each other.
- Solve linear diophantine equations.
- Perform factorization of Fermat numbers.
- Apply the fundamental theorem of arithmetic to find prime factorization of positive integers.
- Use the Euclidean algorithm to produce the greatest common divisor.
- Find the greatest common divisor of two integers.
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Special Congruences and Nonlinear Diophantine Equations
- Determine the sum and number of divisors of positive integers.
- Evaluate the Euler phi-function and solve equations involving the function.
- Use Fermat’s little theorem to find the least positive residue.
- Demonstrate the validity of
- Find the Pythagorean triples.
- Solve some nonlinear diophantine equations as special cases of Fermat's last theorem.
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Congruences
- Solve polynomial congruences.
- Apply the Chinese remainder theorem to solve systems of linear congruences.
- Solve linear congruences.
- Employ modular arithmetic to complete problem sets.
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Integers
- Perform arithmetic operations with integers using algorithms.
- Employ the concept of divisibility of one integer by another integer.
- Implement the Fibonacci sequence.
- Use the principles of mathematical induction to complete positive integer exercises.
- Find sums and products of numbers.
- Verify properties of numbers and sequences.
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Cryptology
- Implement cryptosystems to cryptographic applications.
- Use the RSA cryptosystem to encrypt and decrypt messages.
- Apply the variety of ciphers to encrypt and decrypt messages.
- Perform encryption and decryption on cipher text.
- The University of Phoenix reserves the right to modify courses.
- While widely available, not all programs are available in all locations or in both online and on-campus formats. Please check with a University Enrollment Advisor.
- Transferability of credit is at the discretion of the receiving institution. It is the student’s responsibility to confirm whether or not credits earned at University of Phoenix will be accepted by another institution of the student’s choice.
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