WS2026 Special Topic Lecture in : Algorithmic Number Theory

Time and Place:

Every Mon. 16:15-18:00 (starts Oct. 5, 2026)
JKU T-406/1

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Lecturer:

Jose Capco

Content:

Prime Proving Algorithms (including the deterministic AKS algorithm)
Prime Decomposition Algorithms (including Lenstra's ECM)
Basic Theories on Elliptic Curves over Rationals (e.g. Mordell Theorem)
Elliptic Curve Point Counting Algorithms and Torsion Point Algorithms
Theory of Heights
Elliptic Curve Rank Conjectures (including Birch Swinnerton-Dyer Conjecture)

Course Outline:

On Rings and Groups
Carmichael Numbers and Prime Recognition
Wilson's Theorem and AKS Algorithm
Primitive Root Computations
Quadratic Extensions and Square Root mod p
Sieving B-Smooth Numbers and Basic Quadratic Sieve
Quadratic Sieves. Elliptic Curve Primer
Algorithms on Elliptic Curve Group Structure
Elliptic Curve Factorization Algorithm
Valuation Theory
Computing Heights
Elliptic Curve Rank Conjectures and Elkies Curve
Selected Topics and Algorithms in Algebraic Number Theory
Discussions on Recent Topics in Algorithmic Number Theory

References:

Algorithmische Zahlentheorie, O. Forster, 2. Auflage, 2015 Springer
Prime Numbers, R. Crandall, C. Pomerance, Second Edition, 2005 Springer
A Course in Computational Algebraic Number Theory, H. Cohen, 1996 Springer

Course Requirement:

Elementary Number Theory. Basic understanding of fields, rings and groups.
Open to all students with mathematical background (including computer science students).

Note: First part of the course will have more algorithms than the second part. Main language for the lectures is English. Final grade is based on exercise participation and a presentation of selected recent papers in computational or algorithmic number theory.