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## Number theory(numbertheory)
Traditionally, The term " arithmetic " is also used to refer to number theory. This is asomewhat older term, which is no longer as popular as it once was. Nevertheless, the term remains prevalent --e.g. in the namesof mathematical fields (arithmetic algebraic geometry and the arithmetic of elliptic curves and surfaces). This sense of the termarithmetic should not be confused with the branch of logic which studies arithmetic inthe sense of formal systems.
In Many questions in elementary number theory appear simple but may require very deep consideration and new approaches. Examplesare - The Goldbach conjecture concerning the expressionof even numbers as sums of two primes,
- Catalan's conjecture regarding successive integerpowers,
- The twin prime conjecture about the infinitude of prime pairs , and
- The Collatz conjecture concerning a simple iteration.
The theory of
Diophantine equations
has even been shownto be
In
Many number theoretical questions are best attacked by studying them
Finally, ## History of number theory
The Chebyshev (1850) gave useful bounds for the number of primes between two givenlimits. Riemann (1859) conjectured the limit of the number of primes not exceeding agiven number (the prime number theorem ), introduced complex analysis into the theoryof the Riemann zeta function , and derived the explicit formulae of prime number theory from its zeroes.
The theory of
congruences
may be said to start withGauss's and explored most of the field. Chebyshev published in 1847 a work in Russian on the subject, and in France Serret popularisedit.
Besides summarizing previous work,
Legendre
stated the
law of quadratic reciprocity
. This law, discoveredby induction and enunciated by Euler, was first proved by Legendre in his To Gauss is also due the representation of numbers by binary quadraticforms . Cauchy, Poinsot (1845), Lebesgue (?) (1859, 1868), and notably Hermite have added to thesubject. In the theory of ternary forms Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy advance in the theory of forms in general. Smith gave a completeclassification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. Theinvestigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and thetheory was completed by Smith. Dirichlet was the first to lecture upon the subject in a German university. Among his contributions is the extension ofFermat's theorem on *x*^{n}+*y*^{n}=*z*^{n},
which Euler and Legendre had proved for ## QuotationsMathematics is the queen of the sciences and number theory is the queen of mathematics. Gauss ## References- History of Modern Mathematics by David Eugene Smith, 1906 (adapted public domain text)
*Essays on the Theory of Numbers*, Richard Dedekind, Dover Publications, Inc., 1963. ISBN 0-486-21010-3*Number Theory and Its History*, Oystein Ore, Dover Publications, Inc., 1948,1976. ISBN 0-486-65620-9*Unsolved Problems in Number Theory*, Richard K. Guy, Springer-Verlag, 1981. ISBN 0-387-90593-6 ISBN 3-540-90593-6- Important publications in number theory
number theoyr, integers, number thoery, theorem, unmber theory, euler, numer theory, field, nubmer theory, legendre, numbe rtheory, law, numbertheory, questions, number thory, mathematics, umber theory, smith, numbre theory, arithmetic, numebr theory, quadratic, numbr theory, first, , publications, number theor, fermat, number tehory, reciprocity, number theroy, kummer, nmber theory, dirichlet, number heory, algebra, number theoy, class, nmuber theory, problems, number hteory, many, number thery, factorization, nuber theory, twin, number teory, branch, number theori, statements, numbe theory, work, numbert heory, subject This article is completely or partly from Wikipedia - The Free Online Encyclopedia. Original Article. The text on this site is made available under the terms of the GNU Free Documentation Licence. We take no responsibility for the content, accuracy and use of this article. Anoca.org Encyclopedia 0.01s |