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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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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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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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Number theory is a branch of pure mathematics devoted to the study of the integers and integer-valued functions, exploring properties such as divisibility, prime numbers, and the solutions to equations in integers. It has deep connections with other areas of mathematics and finds applications in cryptography, computer science, and mathematical puzzles.
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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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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.
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Analytic Number Theory is a branch of Number Theory that uses methods from mathematical analysis to solve problems about integers and prime numbers. It is particularly focused on understanding the distribution of prime numbers and employs tools such as complex analysis and infinite series to derive results about number theoretic functions.
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The distribution of prime numbers is a fundamental topic in number theory, revealing how primes become less frequent as numbers grow larger but never completely vanish. This distribution is intricately linked to the Riemann Hypothesis, which suggests a profound connection between the zeros of the Riemann zeta function and the spacing of prime numbers.
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Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves, serving as the building blocks of number theory. They play a crucial role in various fields, including cryptography, due to their properties and distribution patterns.
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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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