- Grades 7-11
- Advanced CTY-Level
Called "The Queen of Mathematics" by the great mathematician Carl Friedrich Gauss, number theory is the study of the natural number system from which all others are derived. Despite the simplicity of the natural numbers, many accessible problems in number theory remain unsolved. For example, the Goldbach Conjecture, formulated in 1742, which posits that every even integer larger than 2 is the sum of two prime numbers, has defied all proof attempts. In this proof-based course, you’ll learn the major ideas of elementary number theory and the historical framework in which they were developed. While strengthening your ability to analyze and construct formal proofs, you and your classmates will explore topics such as the Euclidean Algorithm and continued fractions, Diophantine equations, Fibonacci numbers and the golden ratio, modular arithmetic, Fermat’s Little Theorem, and RSA public key cryptography. You’ll leave the course with an appreciation for the elegance of theoretical mathematics and the ability to craft rigorous arguments.
Typical Class Size: 16-18
- Elaborate, construct, and solve problems of primes using unique factorization, congruences, divisibility, Diophantine equations, primitive roots, and quadratic reciprocity rules
- Elaborate, construct, and solve problems using sums of squares, number-theoretic functions, continued fractions, and apply to public-key encryption, and elliptic curves
- Demonstrate competence with modular arithmetic and apply to Diophantine equations
- Solve Diophantine equations and decide on whether solutions for them exist
- Explain the role of primes, their central role in arithmetic problems and be knowledgeable about primes in number systems other than integers
- Proficient at solving primitive root problems
- Formulate rigorous proofs and defend techniques and theories applied to proofs
Summer Dates & Locations
Testing and Prerequisites
|Required Level||Advanced CTY-Level||Not required|
Students must achieve qualifying scores on an advanced assessment to be eligible for CTY programs. If you don’t have qualifying scores, you have several different testing options. We’ll help you find the right option for your situation.Sign up for Testing Learn More
Course PrerequisitesNumber Theory requires:
Geometry and Algebra 2
Cost and Financial Aid
- Nonrefundable Application Fee - $50 (Waived for financial aid applicants)
- Nonrefundable International Fee - $250 (outside US only)
Financial Aid is available
We are committed to serving all talented youth regardless of financial circumstances. Financial assistance is available based on need.
Please acquire all course materials by the course start date, unless noted as perishable. Items marked as “perishable” should not be acquired until the student needs them in the course. If you have questions about these materials or difficulty locating them, please contact [email protected].
These titles have been featured in past sessions of the course, and may be included this summer. CTY provides students with all texts; no purchase is required.
- Number Theory: A Lively Introduction with Proofs, Applications, and Stories, James Pommersheim, Tim Marks, and Erica Flapan
About Mathematics at CTY
Explore the study of shapes
Many of our courses allow students to describe the world around them in basic and profound ways. Our younger students learn about shape, scale, and proportion in Geometry and Spatial Sense. Middle School students explore beautiful real-world applications of lines; analyze data based on curves that fit a uniform, symmetric and bell-shaped, or skewed pattern in Data and Chance. And advanced students explore the underlying mathematics and fundamental characteristics of shapes, distance, and continuous deformations in our proof-based Topology course.
Dive deep into logic and reasoning
Our courses in formal logic give you the tools to question the world around you. Inductive and Deductive Reasoning introduces younger students to different types of reasoning, as well as the strengths and weaknesses inherent in various forms of critical analysis. Older students explore how logical reasoning can explain (or fail to explain) counter-intuitive results in Paradoxes and Infinities, or take a more rigorous approach to formal logic in Mathematical Logic.