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Topic: Totient function

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	Euler's totient function - Wikipedia, the free encyclopedia
		In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n.
		The totient is usually called the Euler totient or Euler's totient, after the Swiss mathematician Leonhard Euler, who studied it.
		The totient function is important mainly because it gives the size of the multiplicative group of integers modulo n.
	en.wikipedia.org /wiki/Euler's_totient_function (747 words)

	Totient Function -- from MathWorld
		Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function
		The totient function is implemented in Mathematica as
		The totient function is given by the Möbius transform of 1, 2, 3, 4,...
	users.skynet.be /fa956617/math/topics/TotientFunction.html (677 words)

	phi(n) and sigma(n) (Site not responding. Last check: 2007-10-09)
		For any positive integer n, the function div(n) is the number of positive divisors of n, including 1 and n.
		The function φ(n) is the number of elements relatively prime to n.
		The function σ(n) is the sum of all the divisors of n.
	www.mathreference.com /num,phi.html (534 words)

	Static Methods
		Functions are useful when we don't need to remember state from one call to the next.
		Euler's totient function is an important function in number theory: φ(n) is defined as the number of positive integers less than or equal to n that are relatively prime with n (no factors in common with n other than 1).
		The error function is a function that arises in probablity, statistics, and engineering (e.g., solution to differential equations).
	www.cs.princeton.edu /introcs/21function/index.php (4478 words)

	Totient Function (Site not responding. Last check: 2007-10-09)
		Totient is a multiplicative function -- that means when a and b are coprime (i.e.
		Mathworld says that by convention, φ(0) is defined as 1, even though there are no of positive integers that are not larger than 0, so by the definition, φ(0) really should be 0.
		Proof of: For any integer n, the sum of the totient values of each of its divisors equals n.
	mcraeclan.com /mathhelp/BasicNumberCoprimesTotientFunction.htm (732 words)

	Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
		There are 72 residues coprime to 91 (72 is the Euler totient of 91).
		When the 4 factors are prime, (ak+1)(bk+1)(ck+1)(dk+1) is a Carmichael number provided a, b, c and d each divide all of their own symmetric functions: (a+b+c+d), (ab+ac+ad+bc+bd+cd) and (abc+abd+acd+bcd).
		Products of the form (20m+1) (80m+1) (100m+1) (200m+1) are allowed by the sole divisibility of the symmetric functions but m has to be divisible by 3 (m = 3k) or else at least one factor would be so divisible.
	home.att.net /~numericana/answer/modular.htm (3170 words)

	Jiving in J
		The number of totatives of n is called the totient of n, which we may define simply as the length of the list returned by tots.
		When the function on the left is dyadic, the pattern changes: f(g(x)) means x f g(x) in that case, called a hook pattern (there's also a fork pattern).
		The totient of 79 is 78 (is one less than p whenever p is prime).
	www.4dsolutions.net /ocn/Jlang.html (5184 words)

	The Euler function
		is a very important number theoretic function having a deep relationship to prime numbers and the so-called order of integers.
		The Euler function φ: N → N is a mapping associating to each positive integer n the number φ(n) of integers m n relatively prime to n.
		Closely related to the Euler function is the Carmichael function λ.
	www.math-it.de /Mathematik/Zahlentheorie/Euler.html (274 words)

	Arithmetic Functions (Site not responding. Last check: 2007-10-09)
		The divisor function sigma_i(n), which equals the sum of all the d^i for d dividing n, for integer n and small non-negative integer i.
		The function returns 0 if m divides n, -1 if n is not a quadratic residue, and 1 if n is a quadratic residue modulo m.
		This is a multiplicative function characterized by mu(1)=1, mu(p)= - 1, and mu(p^k)=0 for k >= 2, where p is a prime number.
	www.math.colostate.edu /manuals/magma/htmlhelp/text350.html (400 words)

	Marko Riedel's combinatorics and number theory page
		Cyclic groups of functions (f11 z + f12)/(f21 z + f22) under function composition are isomorphic to groups of 2 by 2 matrices [[f11, f12], [f21, f22]] under matrix multiplication.
		Complex residues and the probability that the sum of the values of a roll of n dice is divisible by seven.
		Proof by induction that sum_{dn} d mu(n/d) = phi(n) where phi(n) is Euler's totient function and mu(n) is the Moebius function.
	www.geocities.com /markoriedelde/combnumth.html (1137 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		An arithmetical function is a real- or complex-valued function defined on the integers.
		Properties of these and other functions are widely studied in elementary number theory and recreational mathematics.
		function has numerous applications and is fundamental to the study of congruences.
	www.math.columbia.edu /~rama/chapters/chap15.html (194 words)

	The Half-Totient Tree
		The half-totient function is intimately involved in many interesting areas of mathematics.
		The half-totient function can be used to construct a tree containing all the integers.
		The smallest integer of rank r is often related to Cunningham chains, which give the slowest rate of "descent" under the half-totient function, i.e., decreasing by a factor of 2 on each step.
	www.mathpages.com /home/kmath168.htm (1002 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		From: prezky@apple.com (Michael Press) Subject: Euler's totient function Date: Thu, 12 Oct 2000 16:12:16 -0700 Newsgroups: sci.math Summary: [missing] Paulo Ribenboim's _The Book of Prime Number Records_ is a treat.
		Today I was dazzled by the section on Euler's totient function, and repeat here a few results that particularly captured my imagination.
		Schnizel [1956] There exist infinitely may primes p such that for every k>=1, p.2^k is not a value of Euler's totient function.
	www.math.niu.edu /~rusin/known-math/00_incoming/totient (177 words)

	[No title]
		The inverse modulo function is that number which multiplied by the original number gives one as the remainder.
		EULER'S TOTIENT FUNCTION The Euler totient function, also called the Euler phi function and written as if gcd(x,n) = 1, then x^phi(n) mod n = 1 or x^phi(n) ð 1 mod n phi(n), is the number of positive integers less than n that are relatively prime to n.
		For this reason, Euler's Totient Function is referred to as Euler's generalization of Fermat's little theorem.
	www.und.edu /org/crypto/crypto/lanaki.crypt.class/reports/PKE.TXT (1525 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Euler invented the notion of a totient function.
		The totient of a non-negative integer n, is defined as being the number of non-negative integers less than n which are coprime, or in other words have no common factors with n except 1.
		Classed by some as one of the most important results in mathematics, Euler's identity is a simple consequence of Euler's formula, expressing complex numbers using the exponential function, by letting the angle be pi.
	www.bath.ac.uk /~ech21/euler/accomplish.htm (440 words)

	Prime Number Hide-and-Seek: How the RSA Cipher Works (Site not responding. Last check: 2007-10-09)
		In mathematics, however, a function is used solely for the number it returns.
		In short, we have a recipe for a function that returns its own input (presuming that R has been chosen ahead of time, and that T is verified to be relatively prime to R).
		This, finally, is the heart of what makes RSA a trapdoor function: the gap between obtaining a number with two prime factors, and rediscovering the factors from the number itself.
	www.muppetlabs.com /~breadbox/txt/rsa.html (5933 words)

	Highly totient number - Wikipedia, the free encyclopedia
		The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to φ(x) = k.
		The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient).
		And just as there are infinitely many highly composite numbers, there are also infinitely many highly totient numbers, though the highly totient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into primes, something that becomes extremely difficult as the numbers get larger.
	en.wikipedia.org /wiki/Highly_totient_number (276 words)

	Essays/Totient Function - J Wiki
		The formula readily translates into J: totient=: * -.@%@~.and.q:
		totient 28 12 +/ 1 = 28 +.
		In J6.01, the totient function can also be computed by
	www.jsoftware.com /jwiki/Essays/Totient_Function (52 words)

	Euler Function and Theorem (Site not responding. Last check: 2007-10-09)
		I never saw an authoritative explanation for the name totient he has given the function.
		In Sylvestor's opinion mathematics is essentially about seeing "differences in similarity, similarity in difference." The word totient rhymes with quotient and the function has to do with division but in an unusual way.
		The Euler's totient function for integer m is defined as the number of positive integers not greater than and coprime to m.
	euler-life.org /fun.html (542 words)

	PRIME NUMBER HIDE-AND-SEEK: HOW THE RSA CIPHER WORKS
		In 1975, Whitfield Diffie, Martin E. Hellman, and Ralph Merkle realized that a trapdoor function could be the basis for an entirely new kind of cipher — one in which the decoding method could remain secret even when the encoding method was public knowledge.
		In short, we have a recipe for a function that returns its own input (presuming that R has been chosen ahead of time, and T is relatively prime to R).
		Euler was the first person to publish a proof of this Theorem, and his Totient Theorem is a generalization of Fermat's.
	reactor-core.org /rsa.html (5854 words)

	PlanetMath: Jordan's totient function
		This is a generalization of Euler's Totient Function.
		Cross-references: Euler's totient function, divisors, product, natural numbers, prime
		This is version 8 of Jordan's totient function, born on 2001-08-13, modified 2003-03-30.
	planetmath.org /encyclopedia/JordansTotientFunction.html (38 words)

	Natural to Complex Numbers
		Definition: The Euler totient function phi(n) is defined as the cardinality of the subset of natural numbers strictly less than n, each of which is relatively prime to n.
		Lemma: The totient of a power m of a prime number is phi(p^m) = p^(m - 1) (p - 1).
		Theorem: The totient of n is given by the formula phi(n) = n * product, over primes -- in the semi-closed interval (1, n] -- which divide n, of (1 - 1 / p).
	www.rism.com /Trig/natural_to_complex_numbers.htm (6128 words)

	Relatively prime numbers
		and is called Euler's totient function or phi function.
		Computing the Euler function for a prime modulus p is especially simple.
		function for each one of those prime powers.
	www.math.okstate.edu /~wrightd/crypt/lecnotes/node18.html (296 words)

	The Wedge
		(i) is the Euler totient function defining the number of integers in {1,...,i-1} that are relatively prime to i (with 1 always considered relatively prime to i).
		The latter cardinality is simply the totient function of N [4]:
		The involvement of the totient function in the results for the modified division table suggests a possible connection with the modified multiplication table as well.
	www.balmoralsoftware.com /mathsite/wedge/wedge.htm (1785 words)

	Arithmetic, Numeration, Number Theory - Numericana
		If f is an arithmetic function, a function F, called the sum-function of f, is defined by letting F(n) be the sum of the terms f(d) for all divisors d of n.
		Any arithmetic function f for which f(1) is nonzero has a Dirichlet inverse if we're considering arithmetic functions whose values are in a field.
		A multiplicative function which is zero for squares of primes, and higher powers of prime numbers, is thus the Dirichlet inverse of a totally multiplicative function.
	home.att.net /~numericana/answer/numbers.htm (7607 words)

	The On-Line Encyclopedia of Integer Sequences
		Euler totient function phi(n): count numbers <= n and prime to n.
		W. Sierpinski, Euler's Totient Function And The Theorem Of Euler
		Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
	www.research.att.com /projects/OEIS?Anum=A000010 (574 words)

	Totient Function -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-09)
		, also called Euler's totient function, is defined as the number of positive integers
		The totient function is connected to the Möbius function
		Ruiz, S. "A Congruence With the Euler Totient Function." 11 Oct 2004a.
	mathworld.wolfram.com.cob-web.org:8888 /TotientFunction.html (635 words)

	Index to OEIS (Section To)
		totient function phi(n), does not take these values: A007617*, A005277
		totient function phi(n), iterating: A003434, A007755, A040176, A049108
		totient function phi(n): see also (2): A005239, A006872, A001274, A007015, A001494, A007366, A001837, A001836, A005867
	www.research.att.com /~njas/sequences/Sindx_To.html (177 words)

	smURN - Euler's Totient Function (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-09)
		A prize winning java implementation of Euler's Totient Function φ(n) using the Multiple Polynomial Quadratic Sieve algorithm...
		This year the task was to implement Euler's Totient Function φ(n) which is the number of positive integers smaller n that are relatively prime to n.
		To calculate the totient function the prime factors of n are needed.
	www.smurn.org.cob-web.org:8888 /index.php?option=com_content&task=view&id=26&Itemid=1 (279 words)

	Coprimes and Totient Function (Site not responding. Last check: 2007-10-09)
		The set of coprimes of n, where n is an integer larger than 1, is an infinite set, but considered modulo n, it's a set whose size is the totient of n, φ(n).
		I'll try to put a little notice to that effect at the top of every page.
		Sum of Totients Of Divisors of N is N
	mcraefamily.com /MathHelp/BasicNumberCoprimes.htm (439 words)

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