New Zealand Maths Olympiad Committee online » Number Theory http://www.mathsolympiad.org.nz Mon, 16 Apr 2012 03:34:54 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 Bertrand’s Postulate http://www.mathsolympiad.org.nz/2009/08/bertrands-postulate/ http://www.mathsolympiad.org.nz/2009/08/bertrands-postulate/#comments Tue, 18 Aug 2009 19:43:33 +0000 Michael http://www.mathsolympiad.org.nz/?p=683 The fact that, for every positive integer n, there is a prime between n and 2n is known as Bertrand’s postulate (which is a bit odd, as it’s a theorem, but anyhow …) It arises occasionally in Olympiad style problems (usually with the note “You may assume Bertrand’s Postulate that …”) Michael Nielsen has a nice post giving an elementary proof at the Polymath wiki.

Michael

]]> http://www.mathsolympiad.org.nz/2009/08/bertrands-postulate/feed/ 0 Number theory texts http://www.mathsolympiad.org.nz/2009/02/number-theory-texts/ http://www.mathsolympiad.org.nz/2009/02/number-theory-texts/#comments Tue, 24 Feb 2009 03:26:44 +0000 Heather http://www.mathsolympiad.org.nz/?p=434 (At least!) a couple of good, comprehensive introductions to elementary number theory are available online.  These notes by Jim Hefferon and W. Edwin Clark are nicely written and gently-paced.  These ones by Naoki Sato are a bit more Olympiad-focused.

]]> http://www.mathsolympiad.org.nz/2009/02/number-theory-texts/feed/ 0 Number theory tutorials http://www.mathsolympiad.org.nz/2009/01/number-theory-tutorials/ http://www.mathsolympiad.org.nz/2009/01/number-theory-tutorials/#comments Tue, 20 Jan 2009 10:23:09 +0000 Heather http://www.mathsolympiad.org.nz/?p=296 This series of short introductory articles by Arkadii Slinko covers some of the most fundamental results in number theory.

Tutorial 1: Divisibility and Primes

Tutorial 2: The Euclidean Algorithm

Tutorial 3: Euler’s Function

Tutorial 4: Primes that are Sums of Two Squares

Tutorial 5: Bertrand’s Theorem

(Update, 24/1/09:  some typos fixed.)

]]> http://www.mathsolympiad.org.nz/2009/01/number-theory-tutorials/feed/ 0 Techniques for Diophantine equations http://www.mathsolympiad.org.nz/2009/01/techniques-for-diophantine-equations/ http://www.mathsolympiad.org.nz/2009/01/techniques-for-diophantine-equations/#comments Wed, 14 Jan 2009 08:05:23 +0000 Heather http://www.mathsolympiad.org.nz/?p=258 These notes by Arkadii Slinko outline a number of techniques for solving Diophantine equations.

Solutions for some of the problems are available, and can be obtained by writing to nzmathsolymp@gmail.com.

(Update, 24/1/2009:  some typos fixed.)

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