Prime Numbers Basics

Study Notes

Definition: A prime number is a positive integer that is divisible only by itself and 1.

Key Properties:

Prime numbers are greater than 1.
The only factors of a prime number are 1 and itself.
Prime numbers are always odd, except for 2, which is the only even prime number.

Examples:

Characteristics:

Prime numbers are used as building blocks for all other numbers.
Every positive integer can be expressed as a product of prime numbers in a unique way (Fundamental Theorem of Arithmetic).
Prime numbers are essential in number theory, cryptography, and coding theory.

Types of Prime Numbers:

Mersenne Prime: A prime number that is one less than a power of two (e.g., 3, 7, 31, ...).
Twin Prime: A prime number that is either 2 less or 2 more than another prime number (e.g., 3 and 5, 11 and 13, ...).

Theorems and Conjectures:

Prime Number Theorem: The distribution of prime numbers among the positive integers is approximately uniform.
Riemann Hypothesis: A conjecture about the distribution of prime numbers, which remains unproven.
Goldbach's Conjecture: Every even integer greater than 2 can be expressed as the sum of two prime numbers.

Prime numbers are greater than 1.
The only factors of a prime number are 1 and itself.
Prime numbers are always odd, except for 2, which is the only even prime number.

Prime numbers are used as building blocks for all other numbers.
Every positive integer can be expressed as a product of prime numbers in a unique way (Fundamental Theorem of Arithmetic).
Prime numbers are essential in number theory, cryptography, and coding theory.

A Mersenne prime is a prime number that is one less than a power of two (e.g., 3, 7, 31,...).
A twin prime is a prime number that is either 2 less or 2 more than another prime number (e.g., 3 and 5, 11 and 13,...).

The Prime Number Theorem states that the distribution of prime numbers among the positive integers is approximately uniform.
The Riemann Hypothesis is a conjecture about the distribution of prime numbers, which remains unproven.
Goldbach's Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers.

Roots absorb water and minerals from the soil and anchor the plant in the soil, while also storing food and nutrients.
There are two main types of roots: taproots, which are single, main roots that grow straight down into the soil (e.g., carrots, beets), and fibrous roots, which are networks of smaller, branching roots (e.g., grasses, wheat).
The root structure consists of three main layers: the epidermis, the outermost layer responsible for absorption; the cortex, the inner layer responsible for storage; and the endodermis, the innermost layer responsible for regulating water and mineral flow.

Stems support the plant's leaves, flowers, and fruits, transport water, minerals, and sugars throughout the plant, and store food and nutrients.
The stem structure consists of three main layers: the epidermis, the outermost layer responsible for protection and water retention; the cortex, the inner layer responsible for storage; and the vascular tissue, the innermost layer responsible for transportation.
The vascular tissue is comprised of xylem, which transports water and minerals from roots to leaves, and phloem, which transports sugars and organic compounds from leaves to other parts of the plant.
There are two main types of stems: herbaceous stems, which are soft and non-woody (e.g., grasses, flowers), and woody stems, which are hard and woody (e.g., oak, pine).