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Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

There are approximately 31,444 mathematics articles in Wikipedia.

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Mathematics department in Göttingen where Hilbert worked from 1895 until his retirement in 1930
Image credit: Daniel Schwen

David Hilbert (January 23, 1862, Wehlau, Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation as a great mathematician and scientist by inventing or developing a broad range of ideas, such as invariant theory, the axiomization of geometry, and the notion of Hilbert space, one of the foundations of functional analysis. Hilbert and his students supplied significant portions of the mathematic infrastructure required for quantum mechanics and general relativity. He is one of the founders of proof theory, mathematical logic, and the distinction between mathematics and metamathematics, and warmly defended Cantor's set theory and transfinite numbers. A famous example of his world leadership in mathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century.

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animation of a grid of boxes numbered 2 through 120, where the prime numbers are progressively circled and listed to the side while the composite numbers are struck out
Credit: M.qrius

The sieve of Eratosthenes is a simple algorithm for finding all prime numbers up to a specified maximum value. It works by identifying the prime numbers in increasing order while removing from consideration composite numbers that are multiples of each prime. This animation shows the process of finding all primes no greater than 120. The algorithm begins by identifying 2 as the first prime number and then crossing out every multiple of 2 up to 120. The next available number, 3, is the next prime number, so then every multiple of 3 is crossed out. (In this version of the algorithm, 6 is not crossed out again since it was just identified as a multiple of 2. The same optimization is used for all subsequent steps of the process: given a prime p, only multiples no less than p2 are considered for crossing out, since any lower multiples must already have been identified as multiples of smaller primes. Larger multiples that just happen to already be crossed out—like 12 when considering multiples of 3—are crossed out again, because checking for such duplicates would impose an unnecessary speed penalty on any real-world implementation of the algorithm.) The next remaining number, 5, is the next prime, so its multiples get crossed out (starting with 25); and so on. The process continues until no more composite numbers could possibly be left in the list (i.e., when the square of the next prime exceeds the specified maximum). The remaining numbers (here starting with 11) are all prime. Note that this procedure is easily extended to find primes in any given arithmetic progression. One of several prime number sieves, this ancient algorithm was attributed to the Greek mathematician Eratosthenes (d. c. 194 BCE) by Nicomachus in his first-century (CE) work Introduction to Arithmetic. Other more modern sieves include the sieve of Sundaram (1934) and the sieve of Atkin (2003). The main benefit of sieve methods is the avoidance of costly primality tests (or, conversely, divisibility tests). Their main drawback is their restriction to specific ranges of numbers, which makes this type of method inappropriate for applications requiring very large prime numbers, such as public-key cryptography.

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