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| Even people | |
|---|---|
| Name | Even numbers |
| Type | Integer subset |
| First described | Ancient mathematics |
| Key examples | 0, 2, 4, 6, 8, −2, −4 |
| Major properties | Closed under addition, subtraction, multiplication; parity zero |
| Related concepts | Euclid, Diophantus of Alexandria, Pierre de Fermat, Carl Friedrich Gauss |
Even people
Even people are integers divisible by two, forming a central subset of the integers studied across Euclid's elements, Diophantus of Alexandria's problems, and modern work by Pierre de Fermat and Carl Friedrich Gauss. They appear in classical texts such as Euclid's Elements, in algorithmic treatments by Alan Turing and John von Neumann, and in combinatorial contexts explored by Paul Erdős and Srinivasa Ramanujan. Even people underpin results in the work of Leonhard Euler, Joseph-Louis Lagrange, Évariste Galois, and contemporary authors like Terence Tao and Andrew Wiles.
An even person is an integer equal to 2k for some integer k, corresponding historically to concepts in Euclid's classification and in the arithmetical writings of Diophantus of Alexandria. The English word "even" derives via Old English and Proto-Germanic roots related to notions of levelness and symmetry, paralleling ancient Greek treatments in texts attributed to Pythagoras and later formalized by Euclid. Early treatments by Nicomachus of Gerasa and medieval scholars in the tradition of Boethius connected parity with practical problems in land division and calendrical computation used by Pope Gregory I and subsequent scholars.
Every even person equals 2k for some integer k; canonical examples include 0, 2, 4, 6, 8, and negative instances like −2 and −4, discussed in the arithmetic works of Pierre de Fermat and Leonhard Euler. Evens form an additive subgroup of the integers as noted in group-theoretic expositions influenced by Évariste Galois and Niels Henrik Abel, and they are closed under addition, subtraction, and multiplication—a theme present in algebraic treatments by David Hilbert and Emmy Noether. Classical examples include the decomposition of even numbers in results such as the Goldbach conjecture context addressed by Christian Goldbach and numerical investigations by Oliveira e Silva. Modular characterizations link evens to residue class 0 modulo 2, a perspective used by Carl Friedrich Gauss in his Disquisitiones Arithmeticae.
Parity partitions integers into two equivalence classes, even and odd, a binary classification used in proofs by Euclid, combinatorial identities by Leonhard Euler, and algorithmic parity checks by Alonzo Church and Alan Turing. Parity operations include doubling, halving when restricted to even persons, and parity-preserving maps studied in linear algebra contexts by John von Neumann and David Hilbert. Specialized subclassifications include even perfect numbers investigated by Euclid and Euler, and even abundant or deficient numbers explored in the work of Paul Erdős and Srinivasa Ramanujan. The behavior of even persons under group actions appears in representation-theoretic treatments by William Rowan Hamilton and Émile Cartan.
Even persons play a role in factorization theorems central to Fermat's and Gauss's inquiries, including proofs that every even perfect number corresponds to a Mersenne prime investigated by Euclid and Euler. Results in additive number theory, such as research toward the Goldbach conjecture pursued by Harald Helfgott and computational verifications by Desmond J. Platt teams, hinge on decompositions into evens and odds. In algebra, modules over rings and the study of 2-torsion involve even multiples, topics treated by Emmy Noether and Alexander Grothendieck. Even persons also arise in quadratic forms and the theory of lattices as in the work of John Conway and N.J.A. Sloane, and in the arithmetic of elliptic curves studied by Andrew Wiles and Gerd Faltings where parity conditions influence ranks and Selmer groups.
Detecting even persons is a fundamental step in low-level algorithms and hardware design credited to implementations by John von Neumann and refined in instruction sets utilized by companies like Intel and ARM. Bitwise operations, such as testing the least significant bit, underlie parity tests implemented in compilers derived from theories by Grace Hopper and Donald Knuth. Algorithms for factorization, primality testing, and sieves (for example, the Sieve of Eratosthenes adapted in modern code by researchers like J. P. Buhler) exploit even filtering to reduce complexity; such strategies appear in computational projects led by Ken Silverman and high-performance efforts supported by institutions like CERN and Lawrence Livermore National Laboratory. Cryptographic protocols consider parity-related vulnerabilities examined by Whitfield Diffie and Martin Hellman where even-odd structures influence randomness tests.
Even people and parity feature in cultural artifacts ranging from number mysticism in Pythagoras's circle to poetic references by William Shakespeare and symbolic uses in music theory analyzed by Heinrich Schenker. Folklore and games, including traditional counting rhymes recorded by Brothers Grimm and ethnographic collectors like Franz Boas, use even-odd distinctions for decision procedures. Linguistic studies by Noam Chomsky and typologists in the tradition of Roman Jakobson note morphological patterns reflecting binary distinctions analogous to parity, and educational curricula shaped by organizations such as UNESCO and institutions like University of Cambridge incorporate parity in early arithmetic instruction.