MODERN NUMBER THEORY

Prerequisites & course information

Requisites: MATH 541 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Honors - Honors Optional (%) Repeatable for Credit: No Last Taught: Spring 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in the field of Number Theory (e.g., congruence, quadratic residue, arithmetic function, elliptic curves, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., a number is transcendental, a continued fraction is positive, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of number theory. (e.g., quadratic reciprocity, Roth’s theorem, etc.). Moreover, the student will be able to recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises (e.g., Applying the euclidean algorithm to show that rational numbers have a finite continued fraction expansion , etc.) Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English with appropriate mathematical terminology and notation. Audience: Both Grad & Undergrad 7. Identify applications of course content in areas of modern research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in the field of Number Theory (e.g., congruence, quadratic residue, arithmetic function, elliptic curves, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., a number is transcendental, a continued fraction is positive, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of number theory. (e.g., quadratic reciprocity, Roth’s theorem, etc.). Moreover, the student will be able to recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises (e.g., Applying the euclidean algorithm to show that rational numbers have a finite continued fraction expansion , etc.) Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English with appropriate mathematical terminology and notation. Audience: Both Grad & Undergrad 7. Identify applications of course content in areas of modern research. Audience: Graduate