Number Theory Curriculum

Mathematics

A structured, level-by-level path through Number Theory. Start with Foundations of Divisibility and Core methods in Primes & factorisation at the Intermediate level, then build toward advanced practice and expert-level mastery — with hands-on projects and quizzes at every stage.

3 levels 9 modules 27 topics 6 weeks · about 24 hours · 8 weeks · about 32 hours · 10 weeks · about 40 hours
Intermediate

Intermediate Number Theory

6 weeks · about 24 hours

3 modules

Advanced

Advanced Number Theory

8 weeks · about 32 hours

3 modules

Expert

Expert Number Theory

10 weeks · about 40 hours

3 modules

Number Theory Course Curriculum

9 modules across 3 levels.

Intermediate

Intermediate Number Theory

Start with Divisibility, Primes & factorisation and Modular arithmetic. Each module explains the idea, shows a worked example and ends with a short project, so Number Theory becomes something you can do rather than something you have only read about.

6 weeks · about 24 hours 3 modules

Topics

Project

Write a one-page explanation of Divisibility in your own words, with an example you chose yourself, and test it against the module checklist.

Learning Path

1

Foundations of Divisibility

Work through the three topics in this module, complete the project, then answer the self-check questions before moving on.

2

Core methods in Primes & factorisation

Work through the three topics in this module, complete the project, then answer the self-check questions before moving on.

Prerequisites: Foundations of Divisibility

3

Applying Modular arithmetic

Work through the three topics in this module, complete the project, then answer the self-check questions before moving on.

Prerequisites: Core methods in Primes & factorisation

Practice Problems

EasyExplain Divisibility to someone who has never studied the subject, using one worked example.
Solution: A strong answer defines divisibility precisely, gives one concrete example, and names the limitation it runs into.
MediumExplain Primes & factorisation to someone who has never studied the subject, using one worked example.
Solution: A strong answer defines primes & factorisation precisely, gives one concrete example, and names the limitation it runs into.
HardExplain Modular arithmetic to someone who has never studied the subject, using one worked example.
Solution: A strong answer defines modular arithmetic precisely, gives one concrete example, and names the limitation it runs into.

Glossary

Divisibility

A central idea in the intermediate Number Theory path, covered in the module of the same name.

Example: apply Divisibility to a small, familiar case step by step, then check your reasoning against the walkthrough.

Primes & factorisation

A central idea in the intermediate Number Theory path, covered in the module of the same name.

Example: apply Primes & factorisation to a small, familiar case step by step, then check your reasoning against the walkthrough.

Modular arithmetic

A central idea in the intermediate Number Theory path, covered in the module of the same name.

Example: apply Modular arithmetic to a small, familiar case step by step, then check your reasoning against the walkthrough.

About the Number Theory curriculum

The Number Theory curriculum is a complete, self-paced course in mathematics, organised into 3 progressive levels and 9 modules. It opens with Foundations of Divisibility and Core methods in Primes & factorisation, moves through Applying Modular arithmetic and finishes with expert-level work on Leading practice in Arithmetic functions and Mastering Famous open problems.

Across the whole path you work through 27 individual topics — from Divisibility and Putting divisibility to work at the start to Hands-on: famous open problems and Common pitfalls in famous open problems by the end. The levels are estimated at 6 weeks · about 24 hours, 8 weeks · about 32 hours and 10 weeks · about 40 hours. Each level assumes the one before it, so you can work straight through from Intermediate to Expert, or start at the level that matches what you already know.

It suits analysts, researchers and students who need to reason with numbers and defend a conclusion, not just run a tool. The emphasis is on interpretation: every technique is introduced with the question it answers, the assumption it needs and the way it fails, so you can apply it to unfamiliar data.

Every module pairs a written summary with a worked example, a short quiz to check the idea landed, and a project brief drawn from real practice. You are never asked to take a concept on trust: the reasoning, the common mistakes and the situations where the approach stops working are all laid out, so you finish a module able to explain the idea rather than only recognise it.

What you’ll be able to do

By the end of the Number Theory path you should be able to work confidently with:

Divisibility

Putting divisibility to work

Common pitfalls in divisibility

Primes & factorisation

Hands-on: primes & factorisation

Common pitfalls in primes & factorisation

Modular arithmetic

Putting modular arithmetic to work

How the Number Theory curriculum is structured

Intermediate 6 weeks · about 24 hours 3 modules 9 topics

Intermediate Number Theory

Start with Divisibility, Primes & factorisation and Modular arithmetic. Each module explains the idea, shows a worked example and ends with a short project, so Number Theory becomes something you can do rather than something you have only read about.

Modules: Foundations of Divisibility · Core methods in Primes & factorisation · Applying Modular arithmetic

  • Divisibility: Divisibility is introduced from first principles, with the few terms you actually need and an example that shows how the idea behaves in practice.
  • Putting divisibility to work: Divisibility is where many Number Theory learners first get stuck, so the module slows down, isolates the difficult step and rebuilds it with you.
  • Common pitfalls in divisibility: Divisibility is introduced from first principles, with the few terms you actually need and an example that shows how the idea behaves in practice.

Project: Write a one-page explanation of Divisibility in your own words, with an example you chose yourself, and test it against the module checklist.

Advanced 8 weeks · about 32 hours 3 modules 9 topics

Advanced Number Theory

Build on the intermediate level through Congruences, Diophantine equations and Euler's theorem. The emphasis shifts from knowing the ideas to choosing between them — trade-offs, failure modes and the reasoning behind each decision.

Modules: In depth Congruences · Patterns and trade-offs in Diophantine equations · Scaling Euler's theorem

  • Congruences: The module examines Congruences closely, including the assumptions beneath it and what changes when those assumptions stop holding.
  • Hands-on: congruences: Here Congruences becomes a design decision. You compare alternatives, justify a choice and test it against a harder problem.
  • Checking your understanding of congruences: The module examines Congruences closely, including the assumptions beneath it and what changes when those assumptions stop holding.

Project: Produce a short case analysis in which Congruences drives the recommendation, including the alternatives you rejected and why.

Expert 10 weeks · about 40 hours 3 modules 9 topics

Expert Number Theory

Work at the edge of Number Theory through Cryptographic applications, Arithmetic functions and Famous open problems. The modules assume fluency and ask for judgement: designing, critiquing and defending work at a professional standard.

Modules: Frontiers of Cryptographic applications · Leading practice in Arithmetic functions · Mastering Famous open problems

  • Cryptographic applications: At expert level Cryptographic applications is treated as a field of judgement: current practice, contested positions and the evidence behind each.
  • Hands-on: cryptographic applications: Cryptographic applications is examined at the frontier, with an emphasis on open problems, rigorous argument and decisions you can defend.
  • Common pitfalls in cryptographic applications: At expert level Cryptographic applications is treated as a field of judgement: current practice, contested positions and the evidence behind each.

Project: Write a professional-standard brief on Cryptographic applications — argument, evidence, limitations — plus a plan for taking the work further.

How to study Number Theory, phase by phase

  1. 1

    Phase 1 — Intermediate

    Set aside around 6 weeks · about 24 hours. Work through 3 modules, starting with Foundations of Divisibility and Core methods in Primes & factorisation. Finish the phase by completing the project brief: Write a one-page explanation of Divisibility in your own words, with an example you chose yourself, and test it against the module checklist.

  2. 2

    Phase 2 — Advanced

    Set aside around 8 weeks · about 32 hours. Work through 3 modules, starting with In depth Congruences and Patterns and trade-offs in Diophantine equations. Finish the phase by completing the project brief: Produce a short case analysis in which Congruences drives the recommendation, including the alternatives you rejected and why.

  3. 3

    Phase 3 — Expert

    Set aside around 10 weeks · about 40 hours. Work through 3 modules, starting with Frontiers of Cryptographic applications and Leading practice in Arithmetic functions. Finish the phase by completing the project brief: Write a professional-standard brief on Cryptographic applications — argument, evidence, limitations — plus a plan for taking the work further.

  4. 4

    Phase 4 — Consolidate

    Return to the sections that felt weakest, redo their quizzes from memory, and rebuild the Write a one-page explanation of Divisibility in your own words, with an example you chose yourself, and test it against the module checklist. without following along. Explaining Number Theory to someone else in plain language is the clearest signal that the Expert level has landed.

Once you have finished, the Number Theory path is a reasonable foundation for analytics, data science, research and quantitative roles. Because the curriculum is generated and kept as a structured outline rather than fixed videos, you can revisit any module, regenerate the outline at a different depth, or save it to your library and work through it over weeks rather than in one sitting.

Number Theory — Frequently Asked Questions

Common questions about studying Number Theory on LearnAnyCourse.

What does the Number Theory curriculum cover?

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The Number Theory curriculum is organised into 3 levels (Intermediate, Advanced and Expert) and 9 modules, including Foundations of Divisibility, Core methods in Primes & factorisation, Applying Modular arithmetic, In depth Congruences, Patterns and trade-offs in Diophantine equations and Scaling Euler's theorem.

How long does it take to learn Number Theory?

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Each level is designed to take about 6 weeks · about 24 hours, 8 weeks · about 32 hours and 10 weeks · about 40 hours. Most learners work through one level at a time at their own pace.

Is the Number Theory curriculum free?

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Yes. Every Number Theory curriculum on LearnAnyCourse is free to read and study — create an account only if you want to save curricula to your personal library.

Do I need prior experience to study Number Theory?

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The curriculum starts at the intermediate level and builds upward, so you can begin with the fundamentals and progress to expert-level Number Theory topics without jumping ahead.

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