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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