Number Theory

Course Description

MAS4203 – Number Theory is a 3-credit upper-division mathematics course on the properties of the integers. It is a common requirement in mathematics education programs and an elective for mathematics majors, and it is frequently a student's first sustained experience of writing proofs about objects they already understand.

That combination is what makes number theory pedagogically valuable. The objects are familiar — whole numbers, divisibility, primes — so no effort goes into understanding what is being discussed, and all of it goes into the reasoning. The questions are easy to state and often hard to answer, which gives students genuine mathematical experience: forming a conjecture from computation, testing it, and either proving it or finding why it fails.

Content covers divisibility and the division algorithm; greatest common divisors and the Euclidean algorithm; Bézout's identity and linear Diophantine equations; prime numbers — the fundamental theorem of arithmetic, Euclid's proof of the infinitude of primes, and the distribution of primes; congruences and modular arithmetic; linear congruences and the Chinese remainder theorem; Fermat's little theorem and Euler's theorem; Euler's totient function and other multiplicative functions; Wilson's theorem; primitive roots and the order of an element; quadratic residues and quadratic reciprocity; continued fractions; Diophantine equations including Pythagorean triples; and applications to cryptography, principally RSA.

Offered at approximately 10 Florida institutions offering upper-division mathematics.

Learning Outcomes

Required Outcomes

• Apply the division algorithm and prove basic divisibility results.
• Compute greatest common divisors using the Euclidean algorithm.
• Express a gcd as a linear combination using the extended Euclidean algorithm.
• Solve linear Diophantine equations and characterize all solutions.
• State and apply the fundamental theorem of arithmetic.
• Prove the infinitude of primes and describe results on prime distribution.
• Perform modular arithmetic and prove properties of congruences.
• Solve linear congruences and systems using the Chinese remainder theorem.
• Apply Fermat's little theorem and Euler's theorem to computation and proof.
• Compute Euler's totient function and prove its multiplicativity.
• Apply Wilson's theorem.
• Determine the order of an element and identify primitive roots.
• Determine quadratic residues and evaluate Legendre symbols.
• Apply quadratic reciprocity to decide solvability of quadratic congruences.
• Compute continued fraction expansions and convergents.
• Characterize Pythagorean triples and solve selected Diophantine equations.
• Explain the number-theoretic basis of RSA encryption and perform a small example.
• Construct clear, complete, and correct proofs using induction, contradiction, and direct argument.
• Form a conjecture from computational evidence and test it.

Optional Outcomes

• Describe primality testing and factorization algorithms.
• Describe perfect numbers and Mersenne primes.
• Describe additional cryptographic applications including Diffie-Hellman.
• Use computational tools to explore number-theoretic patterns.
• Describe famous unsolved problems and their history.
• Develop classroom activities based on number-theoretic ideas.

Major Topics

Required Topics

• Divisibility — definitions, properties, and the division algorithm.
• Greatest common divisor — Euclidean and extended Euclidean algorithms.
• Bézout's identity and linear combinations.
• Linear Diophantine equations — solvability and general solution.
• Primes — definition, sieve of Eratosthenes, and infinitude.
• Fundamental theorem of arithmetic — statement, proof, and uniqueness.
• Prime distribution — the prime number theorem stated, and prime gaps.
• Congruences — definition, arithmetic, and residue classes.
• Linear congruences — solvability and modular inverses.
• Chinese remainder theorem — statement, proof, and application.
• Fermat's little theorem — proof and applications.
• Euler's theorem and totient function — multiplicativity and computation.
• Wilson's theorem.
• Order and primitive roots — existence and computation.
• Quadratic residues — Legendre symbol and Euler's criterion.
• Quadratic reciprocity — statement and use.
• Continued fractions — finite and infinite, convergents, and approximation.
• Diophantine equations — Pythagorean triples and selected cases.
• Cryptography — modular exponentiation and the RSA algorithm.
• Proof technique — direct, contradiction, contrapositive, and induction.

Optional Topics

• Primality testing and factorization.
• Perfect numbers and Mersenne primes.
• Diffie-Hellman and further cryptography.
• Computational exploration.
• Unsolved problems and history.
• Classroom applications for future teachers.

Resources & Tools

• Elementary Number Theory and Its Applications (Rosen), Pearson — the most widely adopted text.
• Elementary Number Theory (Burton), McGraw Hill — the other standard; strong historical notes.
• A Friendly Introduction to Number Theory (Silverman) — unusually readable and good at motivating results before proving them.
• Computational tools — Python, SageMath, or Wolfram Alpha; free, and genuinely useful for generating the computational evidence from which conjectures come.
• OEIS (Online Encyclopedia of Integer Sequences) — free; enter a sequence you have computed and discover what is known about it.
• MacTutor History of Mathematics Archive — free biographies and historical context.
• Project Euler — free computational number theory problems; good practice for students who program.

Career Pathways

• Secondary Mathematics Teacher (SOC 25-2031) — the most common destination; requires Florida certification.
• Mathematics Instructor — state college teaching, typically requiring a master's degree.
• Cryptographer and Information Security Analyst (SOC 15-1212) — number theory is the mathematical foundation of public-key cryptography.
• Data Scientist and Quantitative Analyst — where mathematical maturity transfers.
• Software Developer (SOC 15-1252) — algorithmic and cryptographic work.
• Federal technical roles — the National Security Agency is among the largest employers of mathematicians in the country.
• Graduate study — mathematics, computer science, or cryptography.

Special Information

This is a proof course, and that is the point

The single most important expectation to set. Students arriving from a computational calculus sequence often expect problems with numerical answers and find instead that most assignments ask them to prove something. This is where many mathematics students first do real mathematics, and the transition is genuinely hard — not because the material is advanced, but because writing a correct proof is a different skill from executing a procedure. Programs place number theory here deliberately: the objects are elementary enough that the proof technique is the only obstacle.

What helps: read proofs actively, reconstructing each step rather than accepting it; write proofs in complete sentences rather than symbol strings; and get comfortable with not knowing how to start, since struggling before finding the idea is the normal experience rather than a sign of inadequacy.

Compute first, conjecture second, prove third

Number theory rewards experimentation more than most upper-division mathematics, because the objects are computable. Students who work out cases by hand or with a short program — listing residues, computing totients, tabulating quadratic residues — see the patterns that the theorems describe, and arrive at proofs with a sense of why the result should be true. Students who go straight to the proof technique frequently produce correct proofs of statements they do not understand. SageMath and Python are free, and the OEIS will often identify a sequence a student has computed.

RSA makes the abstraction concrete, and it is the course's best hook

Fermat's little theorem, Euler's theorem, and modular exponentiation look like curiosities until they assemble into RSA — the algorithm underlying a substantial share of secure communication. Working a small RSA example by hand, generating keys, encrypting, and decrypting, demonstrates that this ancient-seeming material is load-bearing infrastructure. It also motivates why the difficulty of factoring large integers matters practically, and why quantum computing is discussed as a threat to current cryptography.

The pedagogical value for future teachers is specific

Secondary mathematics teachers encounter divisibility, primes, factorization, GCD and LCM, and modular reasoning throughout the middle and high school curriculum, usually presented as procedures. A teacher who has proved why the fundamental theorem of arithmetic holds, why the Euclidean algorithm terminates, and why divisibility rules work explains those topics with authority rather than assertion. This is a concrete instructional asset, and it is why mathematics education programs require the course rather than treating it as an elective.

Upper-division standing and prerequisites

The 4000-level number means junior or senior standing. Prerequisites normally include the calculus sequence and, importantly, a proof-based course — discrete mathematics, introduction to advanced mathematics, or linear algebra — since the course assumes familiarity with proof technique rather than teaching it from nothing. Students who have not taken such a course should expect a steeper climb. Verify locally. A.A. transfer students should confirm that their receiving university accepts a state college upper-division mathematics course toward the major, since some restrict upper-division transfer or require major coursework in residence. SCNS equivalency applies to the same number at the same level, never across numbers.