Number Theory

What makes prime numbers feel so fundamental compared to other whole numbers? How does factoring a number reveal hidden structure beneath its surface? Why is it much easier to multiply large numbers than to factor them back into primes? How do patterns in prime numbers remain both structured and mysterious at the same time? Can you think of ways that properties of whole numbers show up in puzzles or games you know?

In everyday terms, what does it mean to say that two numbers are the same 'modulo' some base? Where do remainders naturally appear in daily life, such as in clocks, calendars, or seating arrangements? How does modular arithmetic simplify problems that would otherwise be messy with large numbers? Why is this kind of arithmetic so important in modern cryptography and coding? Can you think of a situation where thinking in terms of cycles or wrap-around behavior is helpful?

What kinds of patterns have mathematicians noticed in sequences of numbers like primes or squares? Why are some seemingly simple questions about numbers still unsolved after centuries of work? How do conjectures guide mathematical research even when they remain unproven? What is the difference between noticing a pattern and having a solid proof that it always holds? How might encountering open problems in number theory change a student’s view of math as 'finished'?

How do properties of large primes and factorization underpin many modern encryption systems? Why does the practical difficulty of certain number theory problems help keep digital information secure? What are the risks if advances in algorithms or quantum computing make current cryptography methods easier to break? How should societies balance the desire for strong encryption with concerns about crime or national security? What responsibilities do technologists have when designing systems that rely on mathematical security?

How have different cultures developed distinct systems for writing and working with numbers? What historical problems motivated the study of number theory, long before computers existed? How have religious, philosophical, or mystical views of numbers influenced mathematical thought? What can stories about famous number theorists reveal about the human side of mathematics? Can you think of a number that carries special cultural, religious, or personal meaning?

What does it mean to say that there are infinitely many prime numbers? How can one infinite set of numbers be 'larger' than another, and what does that really imply? Why do ideas about infinity often feel counterintuitive or paradoxical at first? How does thinking carefully about infinite sets sharpen our understanding of the finite world? Can discussions of infinity and number sets influence philosophical or theological ideas about the universe?