2: The Fundamental Theorem of Arithmetic

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2.1: Bézout's Lemma
This page defines key mathematical concepts such as the floor, ceiling, and fractional part of real numbers based on the well-ordering principle. It introduces the norm notation and discusses a lemma on the unique division of real numbers, leading into Theorem 2.5, which presents Bezout's Lemma.
2.2: Corollaries of Be ́zout’s Lemma
This page explores key results in number theory, highlighting Euclid's Lemma, which states that if \(\gcd(a, b) = 1\) and \(a | bc\), then \(a | c\). It derives several theorems, including the Cancellation Theorem, and discusses conditions for multiple divisibility and implications for prime factors in products. The significance of these results is emphasized in the context of solving modular equations and their relevance to understanding divisibility with respect to prime numbers.
2.3: The Fundamental Theorem of Arithmetic
This page covers the definition of units and primes in the integers \(\mathbb{Z}\), identifying \(1\) and \(-1\) as units and distinguishing prime from composite numbers. It presents the Fundamental Theorem of Arithmetic, which states that every nonzero integer has a unique prime factorization, including a unit. The proof employs set construction and the well-ordering principle to show the uniqueness of this factorization through minimality of norms.
2.4: Corollaries of the Fundamental Theorem of Arithmetic
This page covers the unique factorization theorem and its significance in number theory, explaining the relationship between the greatest common divisor and least common multiple with the equation \(\gcd(a, b) \cdot \text{lcm}(a, b) = ab\). It includes a proof involving prime factorization, discusses the infinitude of primes, and concludes with a theorem about the \(k\)-th root of positive integers, alongside a proven lemma regarding the gcd of products.
2.5: The Riemann Hypothesis
This page covers the Riemann zeta function, defined for complex numbers with \(\text{Re} z > 1\) as \(\zeta(z) = \sum_{n=1}^{\infty} n^{-z}\), and its analytic continuation, except at \( z = 1\). It introduces the prime number theorem, which shows that \(\pi(x) \sim \frac{x}{\ln x}\), and discusses the Riemann Hypothesis, asserting that the non-real zeros of \(\zeta(s)\) are on the critical line \(\text{Re}s = \frac{1}{2}\), connecting complex analysis to number theory.
2.6: Exercises
This page contains exercises aimed at enhancing understanding of mathematical concepts such as gcd, lcm, unique factorization, and properties related to prime numbers and the Riemann zeta function. It encourages the application of the division algorithm, provides insights into Euclid's and Bézout’s Lemmas, and examines the divergence of the zeta function while exploring the number theory involving primes.

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