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En estos días estoy leyendo algunos trabajos de Gauss, al que llegué ahora por mi resolución de estudiar teorías gauge. Y una de sus obras, la primera y muy importante, es su Disquisitiones Arithmeticae, donde aborda temas de lo que llama "Aritmética Superior" y lo llamamos teoría de números. Hoy quiero compartir estos enlaces, sobre el tema. En unos días publicaré una segunda tanda, donde incluyo un tema "caliente" en estos días que es la conjetura ABC. Tengo que seguir escribiendo mi serie sobre números primos y completar mi discusión sobre la factorización en factores primos (sólo trabajé con el anillo de enteros). Y alguna vez, quisiera escribir una serie sobre el teorema de Fermat. Tantos temas, una sola vida ;-)
http://en.wikipedia.org/wiki/Number_theory
Number theory (or arithmetic[note 1]) is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).
Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study ofanalytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (diophantine approximation).
Disquisitiones arithmeticae http://books.google.com/books/about/Disquisitiones_arithmeticae.html?id=8LcK_CwzMpQC http://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae
Short Sharp Science: Deceptive puzzle may be solved after 74 years http://www.newscientist.com/blogs/shortsharpscience/2011/06/simple-number-puzzle-possibly.html
Posible demostración de la conjetura de Collatz http://gaussianos.com/posible-demostracion-de-la-conjetura-de-collatz/
I used Collatz sequence as a simple example http://ajlopez.wordpress.com/2010/12/13/azure-multithreads-in-worker-role-an-example/ http://ajlopez.wordpress.com/2010/02/18/agents-in-ajsharp-part-2/
Proizvolov's Identity http://www.cut-the-knot.org/Curriculum/Games/ProizvolovGame.shtml
Números y algo mas...: 719 - Los números brasileños http://simplementenumeros.blogspot.com/2011/06/719-los-numeros-brasilenos.html
Números y algo mas... http://simplementenumeros.blogspot.com/
Fürstenberg's proof of the infinitude of primes http://en.wikipedia.org/wiki/F%C3%BCrstenberg%27s_proof_of_the_infinitude_of_primes
Elon Lindenstrauss and Israel"s Successes in Math http://www.tabletmag.com/news-and-politics/72840/number-theory/
Counting the number of solutions to the Erdös-Straus equation on unit fractions http://terrytao.wordpress.com/2011/07/31/counting-the-number-of-solutions-to-the-erdos-straus-equation-on-unit-fractions/
Fermat's Last Th, n=3 -- Euler's (wrong) proof of Fermat's Last Theorem for n=3 http://2000clicks.com/mathhelp/NumberFermatsLastThCubesEuler.aspx
E241 -- Demonstratio theorematis Fermatiani omnem numerum primum formae 4n+1 esse summam duorum quadratorum http://www.math.dartmouth.edu/~euler/pages/E241.html I found a demostration of this theorem in my last sabbatical week!!!!!! ;-) http://ajlopez.zoomblog.com/archivo/2011/05/21/cinco-al-hilo.html
E98 -- Theorematum quorundam arithmeticorum demonstrationes http://www.math.dartmouth.edu/~euler/pages/E098.html
Euler's four-square identity http://en.wikipedia.org/wiki/Euler%27s_four-square_identity
Sum of Squares Function http://mathworld.wolfram.com/SumofSquaresFunction.html
Fermat's theorem on sums of two squares http://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares
How Euler Did It 26 factors of forms.pdf http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2026%20factors%20of%20forms.pdf
Euler's factorization method http://en.wikipedia.org/wiki/Euler%27s_factorization_method
Brahmagupta–Fibonacci identity http://en.wikipedia.org/wiki/Brahmagupta-Fibonacci_identity
Quadratic reciprocity http://en.wikipedia.org/wiki/Quadratic_reciprocity Ever decade I come back to this topic
Beal's conjecture http://www.bealconjecture.com/
Gauß, Eisenstein, and the ``third'' proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel http://isis.nmsu.edu/~history/schauspiel/schauspiel.html
The Historical Development of the Law of Quadratic Reciprocity http://www.duke.edu/~hdp2/MathHistory2.pdf
Ideal class group http://en.wikipedia.org/wiki/Ideal_class_group
¿Quién no tiene una demostración de la conjetura de Goldbach? http://gaussianos.com/%c2%bfquien-no-tiene-una-demostracion-de-la-conjetura-de-goldbach/
El final de la historia sobre la naturaleza de M67 http://gaussianos.com/el-final-de-la-historia-sobre-la-naturaleza-de-m67/
Dedekind domain http://en.wikipedia.org/wiki/Dedekind_domain
Sphenic number http://en.wikipedia.org/wiki/Sphenic_number
New math theories reveal the nature of numbers http://www.physorg.com/news/2011-01-math-theories-reveal-nature.html
El Teorema de Roth http://www.uam.es/personal_pdi/ciencias/fchamizo/posgrado/masterruiz.pdf
New math theories reveal the nature of numbers http://www.physorg.com/news/2011-01-math-theories-reveal-nature.html
How many numbers are squares mod m http://www.johndcook.com/blog/2008/11/19/how-many-numbers-are-squares-mod-m/
What is the knot associated to a prime? http://www.neverendingbooks.org/index.php/what-is-the-knot-associated-to-a-prime.html Knots, another topic from my last sabbatical week. I should write a post! ;-)
Pell's equation http://www-history.mcs.st-and.ac.uk/HistTopics/Pell.html
Solvin the Pell Equation http://www.ams.org/notices/200202/fea-lenstra.pdf
Pell's equation http://www.ieeta.pt/~tos/pell.html
Amazon.com: Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and its Applications) Valérie Berthé, Michel Rigo: Books http://www.amazon.com/Combinatorics-Automata-Encyclopedia-Mathematics-Applications/dp/0521515971
Algunas fracciones continuas interesantes | Gaussianos http://gaussianos.com/algunas-fracciones-continuas-interesantes/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+gaussianos+%28Gaussianos%29
The Ulam Spiral https://bitbucket.org/ctkrohn/ulam/wiki/Home
Fracción continua, o cuál es la mejor aproximación http://gaussianos.com/fraccion-continua-o-cual-es-la-mejor-aproximacion/
La cuadratura del rectangulo http://gaussianos.com/la-cuadratura-del-rectangulo/
Deterministic methods to find primes « What"s new http://terrytao.wordpress.com/2010/09/21/a-deterministic-way-to-find-primes/
Perfect number http://en.wikipedia.org/wiki/Perfect_number This topic was one of my first encounter with mathematics
No odd perfect number found yet, but: Un número perfecto impar debe tener al menos tres factores primos http://gaussianos.com/un-numero-perfecto-impar-debe-tener-al-menos-tres-factores-primos/
Ecuación en enteros http://gaussianos.com/ecuacion-en-enteros/
El número plateado http://gaussianos.com/el-numero-plateado/
196 and Other Lychrel Numbers http://www.p196.org/
Lychrel number http://en.wikipedia.org/wiki/Lychrel_number
Fundamental groups and Diophantine geometry http://www.ucl.ac.uk/~ucahmki/leeds.pdf Group theory, another fascinating topic
Kim on Fundamental Groups in Number Theory | The n-Category Cafe http://golem.ph.utexas.edu/category/2008/03/kim_on_fundamental_groups_in_n.html
The music of primes http://www.musicoftheprimes.com/index.htm
Euler's Totient Function http://uva.onlinejudge.org/external/110/11073.html
Grytczuk, Wójtowicz: On a Lehmer problem concerning Euler's totient function http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pja/1116443716
Totient Function http://mathworld.wolfram.com/TotientFunction.html
Mis post sobre la función indicatriz de Euler ("totient function") http://ajlopez.zoomblog.com/archivo/2010/06/30/la-funcion-indicatriz-de-Euler-primero.html http://ajlopez.zoomblog.com/archivo/2010/07/19/una-propiedad-de-la-indicatriz-de-Eule.html
A remark on partial sums involving the Möbius function http://terrytao.wordpress.com/2009/08/30/an-elementary-inequality-involving-the-mobius-function/
The Surprises Never Eend: The Ulam Spiral of Primes : Good Math, Bad Math http://scienceblogs.com/goodmath/2010/06/the_surprises_never_eend_the_u.php
Lattices and their invariants « Secret Blogging Seminar http://sbseminar.wordpress.com/2010/05/14/lattices-and-their-invariants/
Marker lecture III: "Small gaps between primes" http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/
Finding primes - Polymath1Wiki http://michaelnielsen.org/polymath1/index.php?title=Finding_primes
Mis Enlaces http://www.delicious.com/ajlopez/numbertheory
Nos leemos!
Angel "Java" Lopez http://www.ajlopez.com http://twitter.com/ajlopez
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