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Proof that is irrational
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71	IRRATIONAL NUMBERS ARISING FROM CERTAIN DIFFERENTIAL EQUATIONS M. RAM MURTY AND V. KUMAR MURTY Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven’s proof to show that er is irrational for ev Add to Reading List Source URL: rinconmatematico.com Language: English - Date: 2005-04-23 05:41:16 Irrational number Integer sequences Mathematical analysis Transcendental number Euclidean algorithm Mathematics Pi Proof that π is irrational
72	Proceedings of the Edinburgh Mathematical Society[removed], 271–289 DOI:[removed]S0013091510001355 ON NUMBERS n DIVIDING THE nTH TERM OF A LINEAR RECURRENCE 1 Add to Reading List Source URL: www.math.dartmouth.edu Language: English - Date: 2012-05-10 10:36:06 Modular arithmetic Arithmetic function Prime-counting function Exponentiation Fibonacci number Binomial coefficient Proof that π is irrational Mathematics Number theory Prime numbers
73	UIUC Mock Putnam Exam[removed]Solutions Problem 1. Without any numerical calculations, determine which of the two numbers eπ and π e is larger. Solution. Let a = eπ and b = π e . We will show that a > b. Since Add to Reading List Source URL: www.math.illinois.edu Language: English - Date: 2005-10-11 15:50:00 Integer sequences Polynomial Exponentiation Logarithm Pi Factorial Proof that π is irrational Binomial coefficient Mathematics Mathematical analysis Combinatorics
74	The Gift Exchange Problem David Applegate and N. J. A. Sloane(a) , AT&T Shannon Labs, 180 Park Ave., Florham Park, NJ[removed], USA. (a) Add to Reading List Source URL: neilsloane.com Language: English - Date: 2012-01-02 21:20:57 Number theory Orthogonal polynomials Bessel polynomials Summation Factorial Recurrence relation Combinatory logic Stirling number Proof that π is irrational Mathematics Integer sequences Combinatorics
75	2011 U OF I FRESHMAN MATH CONTEST Solutions 1. Let x = 0.[removed][removed]be the number whose decimal expansion consists of the sequence of natural numbers written next to each other. (a) Determine the 2011th digi Add to Reading List Source URL: www.math.illinois.edu Language: English - Date: 2011-10-01 17:22:28 Combinatorics Number theory Mathematical series Complex analysis Binomial coefficient Factorial Pi Summation Proof that π is irrational Mathematics Mathematical analysis Integer sequences
76	UIUC UNDERGRADUATE MATH CONTEST April 16, 2005 Solutions 1. For which positive integers n does the equation a1 a2 + a2 a3 + · · · + an−1 an + an a1 = 0 Add to Reading List Source URL: www.math.illinois.edu Language: English - Date: 2005-04-16 17:33:25 Trigonometry Logarithms Exponentiation Sine Sinc function Mathematical fallacy Proof that π is irrational Fourier series Mathematics Special functions Mathematical analysis
77	2014 UI UNDERGRADUATE MATH CONTEST Solutions 1. (a) Does there exist a multiple of 2013 whose decimal representation ends in the digits 2014? Explain! (b) Does there exist a power of 2 whose decimal representation ends i Add to Reading List Source URL: www.math.illinois.edu Language: English - Date: 2014-02-15 21:13:47 Integer sequences Number theory Exponentials Binomial coefficient Factorial Summation Exponential function Proof that π is irrational Basel problem Mathematics Mathematical analysis Combinatorics
78	UIUC UNDERGRADUATE MATH CONTEST April 12, 2003 Solutions 1. Let Add to Reading List Source URL: www.math.illinois.edu Language: English - Date: 2003-04-12 16:50:35 Integer sequences Number theory Factorial Proof that π is irrational Binomial coefficient Mathematics Mathematical analysis Combinatorics
79	U OF I UNDERGRADUATE MATH CONTEST MARCH 7, 2009 Solutions − denote the integer obtained by reversing the 1. Given a nonnegative integer n, let ← n Add to Reading List Source URL: www.math.illinois.edu Language: English - Date: 2009-03-07 17:49:43 Combinatorics Exponentials Factorial Exponentiation Binomial coefficient Proof that π is irrational Mathematics Integer sequences Number theory
80	Ballinteer Institute © [removed]www.leavingcertsolutions.com Leaving Cert Higher Maths Sequences and Series Add to Reading List Source URL: www.leavingcertsolutions.com Language: English - Date: 2004-02-29 11:04:45 Complex analysis Real analysis Taylor series Summation Power series Series Bernoulli polynomials Proof that π is irrational Mathematical analysis Mathematics Mathematical series

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