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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are fundamental in number theory due to their role as the building blocks of the integers, analogous to atoms in chemistry.
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Divisibility is a fundamental concept in number theory that determines whether one integer can be divided by another without leaving a remainder. Understanding divisibility helps in simplifying fractions, finding greatest common divisors, and solving problems involving modular arithmetic.
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Prime factorization is the process of expressing a number as the product of its prime factors, which are the prime numbers that multiply together to yield the original number. This is a fundamental concept in number theory, crucial for understanding the properties of numbers, solving problems in arithmetic, and applications in cryptography.
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Composite numbers are positive integers that have more than two distinct positive divisors, meaning they can be divided evenly by numbers other than 1 and themselves. They are the opposite of prime numbers and start from 4, as 2 and 3 are prime numbers.
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The Euclidean Algorithm is an efficient method for computing the greatest common divisor (GCD) of two integers, which works by repeatedly applying the division algorithm and reducing the problem size. It is fundamental in number theory and forms the basis for more advanced algorithms in computational mathematics.
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The Sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to a specified integer by iteratively marking the multiples of each prime number starting from 2. It efficiently eliminates non-prime numbers in a range by progressively marking the multiples of each prime, resulting in a list of primes up to the desired limit.
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers, up to the order of the factors. This theorem establishes the essential role of prime numbers as the 'building blocks' of the integers and underpins much of number theory.
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Goldbach's Conjecture is an unproven hypothesis in number theory that suggests every even integer greater than two can be expressed as the sum of two prime numbers. Despite extensive computational evidence supporting it, a formal proof has yet to be discovered, making it one of the oldest unsolved problems in mathematics.
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Twin primes are pairs of prime numbers that have a difference of two, such as (3, 5) or (11, 13). The Twin Prime Conjecture, an unsolved problem in number theory, posits that there are infinitely many Twin primes, although this has not been proven yet.
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Prime gaps refer to the differences between consecutive prime numbers, which can vary widely despite the infinite distribution of primes. Understanding Prime gaps is crucial in number theory as they provide insights into the distribution of primes and have implications for unsolved problems like the Twin Prime Conjecture and the distribution of prime numbers in arithmetic progressions.
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Mersenne primes are a special class of prime numbers that can be expressed in the form 2^p - 1, where p itself is a prime number. They are significant in number theory and cryptography due to their properties and the role they play in the search for large prime numbers.
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Euclid's Theorem states that there are infinitely many prime numbers, establishing that no finite list can encompass all primes. This foundational result in number theory was proven by Euclid around 300 BCE, using a method that involves constructing a new prime from any given list of primes.
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Fermat primes are a special class of prime numbers that can be expressed in the form 2^(2^n) + 1, where n is a non-negative integer. Only five Fermat primes are currently known, and it is an open question whether there are any more beyond these five.
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A primality test is an algorithm used to determine whether a given number is prime, which is crucial in fields like cryptography and number theory. Efficient primality testing is essential for secure encryption systems, as it enables the identification of large Prime Numbers necessary for cryptographic keys.
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Prime Number Theory is a branch of mathematics focused on understanding the distribution and properties of prime numbers, which are numbers greater than 1 that have no divisors other than 1 and themselves. It plays a crucial role in Number Theory and has significant applications in cryptography, computer science, and mathematical research.
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The Twin Prime Conjecture posits that there are infinitely many pairs of prime numbers, known as twin primes, that have a difference of two, such as ((3, 5)) and ((11, 13)). Despite significant progress in number theory, the conjecture remains unproven, although recent advancements have shown there are infinitely many pairs of primes with bounded gaps, moving closer to a potential proof.
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The infinitude of primes is a fundamental theorem in number theory, stating that there are infinitely many prime numbers. This was first proven by Euclid, who showed that for any finite list of primes, there is always another prime not on the list, thus ensuring the endless continuation of primes.
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Quadratic Reciprocity is a fundamental theorem in number theory that provides criteria to determine the solvability of quadratic equations modulo prime numbers. It reveals a surprising and profound symmetry in the behavior of quadratic residues, linking the solvability of one prime modulo another to the reverse scenario.
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Fermat's Little Theorem states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p). This theorem is fundamental in number theory and has applications in fields such as cryptography, particularly in algorithms like RSA encryption.
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Primality testing is the process of determining whether a given number is prime, which is crucial for cryptographic applications and number theory. Efficient primality tests, such as the AKS primality test, have been developed to handle large numbers, providing deterministic results in polynomial time.
Dirichlet's Theorem on Arithmetic Progressions states that for any two coprime integers a and d, there are infinitely many prime numbers of the form a + nd, where n is a non-negative integer. This theorem highlights the distribution of prime numbers in arithmetic sequences and is foundational in analytic number theory.
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Prime numbers are special numbers that can only be divided by 1 and themselves without leaving any leftovers. They are like the building blocks of all other numbers because every number can be made by multiplying prime numbers together.
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The Green-Tao Theorem shows that you can always find a pattern of numbers that add up by the same amount, like counting by twos or threes, even if you only use prime numbers. prime numbers are special because they can only be divided by 1 and themselves without leaving anything left over.
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