Number Theory Explorations

Number theory is the queen of mathematics, and this book explores three of its most fascinating corners: Fermat numbers, Euler’s prime-generating polynomial, and the Collatz conjecture.

The Fermat numbers chapter covers why the exponent must be a power of 2, Euler’s factorization of F_5, Bennett’s elegant proof that 641 divides F_5, mutual coprimality, the Euler-Lucas factoring theorem, Pepin’s primality test, and connections to constructible polygons.

Euler’s formula n^2 + n + 41 generates primes for n = 0 to 39. The book explains why it fails at n = 40, explores the connection to Heegner numbers, identifies the Lucky Numbers of Euler, and proves why no polynomial can generate all primes. The Ulam spiral connection adds a visual dimension.

The Collatz conjecture section takes an honest approach. It presents a sieve-based argument, then systematically critiques why it is insufficient for a proof. The chapter covers computational verification, partial results, possible failure modes, and why the problem remains so hard.

Key Topics

• Fermat numbers: properties, primality testing, and factorization
• Euler’s prime-generating polynomial and Heegner numbers
• The Collatz conjecture: evidence, approaches, and obstacles
• Connections between the three topics via quadratic residues
• Complete tables of Fermat factorizations and Euler formula values
• Open problems and computational frontiers