Primality Tests and Prime Number Theorem

Trial Division: Check if a number is prime by dividing it by all prime numbers less than or equal to its square root.
Lucas-Lehmer Primality Test: Used to test Mersenne numbers for primality.
AKS Primality Test: A polynomial-time algorithm to determine whether a given number is prime or composite.
Miller-Rabin Primality Test: A probabilistic algorithm to test whether a number is prime or composite.

Statement: The prime number theorem (PNT) describes the distribution of prime numbers among the positive integers.
Approximation: The number of prime numbers less than or equal to x, denoted by π(x), is approximately equal to x / ln(x) as x approaches infinity.
Implications: The PNT has far-reaching implications in many areas of mathematics, including number theory, algebra, and analysis.

Definition: Twin primes are pairs of prime numbers that differ by 2, such as (3, 5) and (11, 13).
Conjecture: The twin prime conjecture states that there are infinitely many twin primes.
Recent developments: In 2013, Yitang Zhang made significant progress towards proving the conjecture, showing that there are infinitely many pairs of primes differing by at most 70 million.

Definition: A Mersenne prime is a prime number that can be written in the form M_n = 2^n - 1, where n is also a prime number.
Properties: Mersenne primes have the form 2^p - 1, where p is also a prime number.
Examples: 3, 7, 31, and 127 are all Mersenne primes.

RSA Algorithm: A public-key encryption algorithm that relies on the difficulty of factoring large composite numbers into their prime factors.
Security: The security of RSA is based on the hardness of factoring large composite numbers, which is closely related to the distribution of prime numbers.
Prime numbers in cryptography: Prime numbers play a crucial role in many cryptographic protocols, including Diffie-Hellman key exchange and elliptic curve cryptography.

Trial Division is a method to check if a number is prime by dividing it by all prime numbers less than or equal to its square root.
Lucas-Lehmer Primality Test is used to test Mersenne numbers for primality.
AKS Primality Test is a polynomial-time algorithm to determine whether a given number is prime or composite.
Miller-Rabin Primality Test is a probabilistic algorithm to test whether a number is prime or composite.

The Prime Number Theorem (PNT) describes the distribution of prime numbers among the positive integers.
The number of prime numbers less than or equal to x, denoted by π(x), is approximately equal to x / ln(x) as x approaches infinity.
The PNT has far-reaching implications in many areas of mathematics, including number theory, algebra, and analysis.

Twin primes are pairs of prime numbers that differ by 2, such as (3, 5) and (11, 13).
The twin prime conjecture states that there are infinitely many twin primes.
In 2013, Yitang Zhang made significant progress towards proving the conjecture, showing that there are infinitely many pairs of primes differing by at most 70 million.

A Mersenne prime is a prime number that can be written in the form M_n = 2^n - 1, where n is also a prime number.
Mersenne primes have the form 2^p - 1, where p is also a prime number.
Examples of Mersenne primes include 3, 7, 31, and 127.

The RSA Algorithm is a public-key encryption algorithm that relies on the difficulty of factoring large composite numbers into their prime factors.
The security of RSA is based on the hardness of factoring large composite numbers, which is closely related to the distribution of prime numbers.
Prime numbers play a crucial role in many cryptographic protocols, including Diffie-Hellman key exchange and elliptic curve cryptography.

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