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Topic: Euler pseudoprime

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Euler characteristic

Euler angles

Euler Mascheroni constant

Euler equations

Euler Lagrange equations

Euler numbers

Euler Maclaurin formula

Euler product

Euler Jacobi pseudoprime

Leonard Euler

Euler pseudoprime

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	Euler pseudoprime
		An odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and
		Every Euler pseudoprime is also a Fermat pseudoprime.
		It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.
	www.ebroadcast.com.au /lookup/encyclopedia/eu/Euler_pseudoprime.html (239 words)

	Euler pseudoprime: Definition and Links by Encyclopedian.com (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		It is not possible to produce a definite test of primality based on whether a number is an Euler...
		there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base...The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or...
		The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 561 = 3·11·17.
	www.encyclopedian.com.cob-web.org:8888 /eu/Euler-pseudoprime.html (447 words)

	Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality - Numericana
		The most studied pseudoprimes are pseudoprimes to base 2, which have been variously called Poulet numbers, Fermatians, or Sarrus numbers...
		Conversely, a weak pseudoprime that's coprime with the base is a pseudoprime in the usual sense, otherwise this may or may not be the case.
		We may observe that 91 is thus coprime to twice as many bases as it's a pseudoprime to (72 is the Euler totient of 91).
	home.att.net /~numericana/answer/pseudo.htm (3003 words)

	Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
		91 is a pseudoprime to half of the bases coprime to itself.
		There are 72 residues coprime to 91 (72 is the Euler totient of 91).
		A Carmichael number is thus a pseudoprime to any base a to which it's coprime.
	home.att.net /~numericana/answer/modular.htm (3170 words)

	Euler-Jacobi pseudoprime
		Every Euler-Jacobi pseudoprime is also a Fermat pseudoprime and an Euler pseudoprime.
		There are no numbers which are Euler-Jacobi pseudoprimes to all bases as Carmichael numbers are.
		The table below gives all Euler-Jacobi pseudoprimes less than 10000 for some prime bases a, this table is in the process of being checked and should be used with caution until this notice is removed.
	ebroadcast.com.au /lookup/encyclopedia/eu/Euler-Jacobi_pseudoprime.html (387 words)

	Number Theory Glossary
		A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases.
		Also called the Euler pseudoprime test, this test was originally proposed by Solovay and Strassen in SIAM J. Computing, 6 (1977), 84-85 and 7 (1978), 118.
		If an integer is a strong pseudoprime it is also a Fermat pseudoprime and an Euler pseudoprime.
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (827 words)

	Strong pseudoprime (Site not responding. Last check: 2007-11-03)
		In mathematics, a strong pseudoprime is a certain kind of natural number.
		A strong pseudoprime to base a is always an Euler pseudoprime to base a (Pomerance, Selfridge, Wagstaff 1980), but not all Euler pseudoprimes are strong pseudoprimes.
		As Monier and Rabin showed in 1980, a composite number n is a strong pseudoprime to at most one quarter of all bases Carmichael numbers", numbers that are strong pseudoprimes to all bases.
	www.myfasthomeequityloan.com /wiki/?title=Strong_pseudoprime (212 words)

	Euler-Jacobi pseudoprime - Wikipedia, the free encyclopedia
		In number theory, an odd composite integer n is called an Euler-Jacobi pseudoprime to base a, if a and n are coprime, and
		The motivation for this definition is the fact that all prime numbers n satisfy the above equation, as explained in the Legendre symbol article.
		These numbers are, in some sources, called Euler pseudoprimes.
	en.wikipedia.org /wiki/Euler-Jacobi_pseudoprime (434 words)

	Pseudoprime -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		A pseudoprime is a composite number that passes a test or sequence of tests that fail for most
		The following table gives the number of Poulet numbers psp(2), Euler-Jacobi pseudoprimes ejpsp(2), and strong pseudoprimes spsp(2) to the base 2, and Carmichael numbers CN that are smaller than the first few powers of 10 (Guy 1994).
		Fermat Pseudoprime, Fibonacci Pseudoprime, Frobenius Pseudoprime, Lucas Pseudoprime, Perrin Pseudoprime,
	mathworld.wolfram.com.cob-web.org:8888 /Pseudoprime.html (233 words)

	Math 5410 Midterm Exam
		(b) Prove that any Euler pseudoprime to the base b is a pseudoprime to the base b.
		- 1 is a strong pseudoprime and an Euler pseudoprime to the base 2.
		This is a cyclic shift cipher by 13 (i.e.
	www-math.cudenver.edu /~wcherowi/courses/m5410/m5410mid.html (969 words)

	Section 7
		For the two types of pseudoprime already mentioned, there is no measure of the probability of composite n surviving repeated tests; we just accept that it decreases substantially at each iteration.
		Every strong pseudoprime to the base a is already an Euler pseudoprime to the base a.
		It is better to use the repeat pseudoprime method for a smaller range of test values, and using the fact that the probability of a composite n passing each iteration of the test is at most 1/4, so that after k successful tests, the probability that n is prime is greater than 1
	www.glasgowg43.freeserve.co.uk /pfaq7.htm (2603 words)

	Encyclopedia Search
		pseudoprime An odd composite integer n is...is called an Euler-
		pseudoprime to base a, if a and n are coprime, and a (n -1)/2 = (a /...
		symbol, is sometimes used for a definition of an Euler pseudoprime.
	www.encyclopedian.com /search.php?searWords=Jacobi (133 words)

	Earliest Known Uses of Some of the Words of Mathematics (E) (Site not responding. Last check: 2007-11-03)
		The conjecture was shown to be false by Bose, Shrikhande, and Parker in 1959.
		EULER'S NUMBERS (for the coefficients of a series for the secant function) were so named by H. Scherk in 1825 in Vier mathematische Abhandlungen (Cajori vol.
		The OED2’s earliest quotation for Euler's theorem is from 1847 in Phil.
	members.aol.com /jeff570/e.html (8885 words)

	Euler-Jacobi pseudoprime: Definition and Links by Encyclopedian.com (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		...Euler-Jacobi pseudoprime Euler-Jacobi pseudoprime An odd composite integer n is...n is called an Euler-Jacobi pseudoprime to base a, if a and n are coprime, and a (n -1)/2 =...to base a, if a and n are coprime, and a (n -1)/2 = (a / n) (mod n), where (a / n) is the...
		...prime which is composite is called an Euler-Jacobi pseudoprime.
		Probable primes find application in cryptography....also: Pseudoprime Euler-Jacobi pseudoprime Carmichael number nties exist in the nature of reality...
	www.encyclopedian.com.cob-web.org:8888 /eu/Euler-Jacobi-pseudoprime.html (568 words)

	Math 208A (Site not responding. Last check: 2007-11-03)
		Show that 1105 is a Carmichael number and find the smallest b>1 such that 1105 is not an Euler pseudoprime to the base b.
		Find the smallest b such that 2821 is not a strong pseudoprime to the base b.
		Explain why a base b to which n is not a strong pseudoprime can be used to find a non-trivial factor of n.
	www.math.ucla.edu /~blasius/208a.1.01f/math208hw.htm (256 words)

	Chapter8.htm
		Proof: The theorem is true for any finite group, but we will prove it only for the abelian case.
		A Fermat pseudoprime base b is a composite number
		A Carmichael number is a pseudoprime for every possible base
	www.math.ups.edu /~riegsecker/CSCI425/Chapter8/Chapter8.htm (222 words)

	The Prime Glossary: Euler probable prime (Site not responding. Last check: 2007-11-03)
		is one modulo p. Euler was able to prove the stronger statement: a
		If this is the case and n is composite, then we say n is an Euler pseudoprime (base a).
		Below is a table of the odd composite Euler PRP's less than 3000 and the percentage of the odd composites in this range that are not exposed as composite by this test.
	primes.utm.edu /glossary/page.php?sort=EulerPRP (240 words)

	Pseudoprime (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		Composite'' requirement, calling any number which passes the specified tests a pseudoprime even if it is
		Composite numbers which are pseudoprimes to every base; they are sometimes called
		Strong Pseudoprimes spsp to the base 2, as well as
	bbs.sachina.pku.edu.cn.cob-web.org:8888 /Stat/Math_World/math/p/p715.htm (88 words)

	Amazon.com: "odd pseudoprime": Key Phrase page (Site not responding. Last check: 2007-11-03)
		Indeed we show that if n is an odd pseudoprime, then so is m = 2" - 1.
		This construction of pseudoprimes can be continued ad infinitum.
		We call an integer n, which is composite and odd, pseudoprime with respect to a if (a, n) = 1 and a"-' = 1 (mod n).
	www.amazon.com /phrase/odd-pseudoprime (402 words)

	pseudoprime - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "pseudoprime" is defined.
		Pseudoprime : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include pseudoprime: frobenius pseudoprime, perrin pseudoprime, euler jacobi pseudoprime, fibonacci pseudoprime, somer-lucas pseudoprime, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=pseudoprime (102 words)

	IBigInteger Members (Site not responding. Last check: 2007-11-03)
		True if "this" is a pseudoprime to randomly chosen bases.
		True if "this" is a strong pseudoprime to randomly chosen bases.
		Sets the value of the specified bit to 1 The Least Significant Bit position is 0.
	www.pivo.com /doc/com/Pivo.SSH.IBigIntegerMembers.html (294 words)

	Euler-Jacobi Pseudoprime (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		Composite number is an Euler-Jacobi pseudoprime for all bases
		An Euler pseudoprime is pseudoprime to at most 1/2 of all possible bases less than itself.
		The first few base-2 Euler-Jacobi pseudoprimes are 561, 1105, 1729, 1905, 2047, 2465,...
	www.math.sdu.edu.cn.cob-web.org:8888 /mathency/math/e/e309.htm (71 words)

	Amazon.com: "Euler's Proposition": Key Phrase page (Site not responding. Last check: 2007-11-03)
		Descartes on Polyhedra: A Study of the "De solidorum elementis" (Sources in the History of Mathematics and Physical Sciences) by P. Federico
		See all pages with references to Euler's Proposition.
		the sum of the plane angles in terms of the number of faces and plane angles, finds its counterpart in Euler's Proposition VIII, which gives this sum in terms of the number of faces and edges.
	www.amazon.com /phrase/Euler's-Proposition (437 words)

	Cryptography Theory and Practice
		Page 132: Lines -16 and -15 should read as follows: "If this equation holds, then n is called an Euler pseudo-prime to the base a.
		For example, 91 is an Euler pseudo-prime to the base 10...".
		Page 160: Exercise 4.13 should read as follows: "For n = 837, 851 and 1189, find the number of bases b such that n is an Euler pseudoprime to the base b".
	www.cacr.math.uwaterloo.ca /~dstinson/errata.html (2319 words)

	Math 373
		I'll be using the book by Stan Wagon and myself: A Course in Computational Number Theory.
		We'll cover the standard topics: the Euclidean algorithm, modular arithmetic, linear congruences, Chinese remainder theorem, Fermat's Little Theorem, primality testing using pseudoprime tests, Euler's phi function, perfect numbers, Euler's theorem, primitive roots, the distribution of prime numbers, how to certify that a number is prime, RSA cryptosystems, check digit schemes, factoring algorithms, and quadratic residues.
		Throughout the course, we will be using the computer (using Mathematica programs that will be supplied) to explore patterns in the integers, use those explorations to guess what is happening and motivate the theory, and then use the theory to show what to explore next.
	www.macalester.edu /~bressoud/courses/math373.html (644 words)

	NUMBER THEORY AND CRYPTOGRAPHY. Assignments
		(Note that if n is a pseudoprime to the base b, and n is a pseudoprime to the base c, then n is a pseudoprime to the base bc.
		Make tables showing all the quadratic residues and nonresidues modulo 23 and modulo 29.
		Find a number b so that 561 is an Euler pseudoprime to the base b but 561 is not a strong pseudoprime to the base b.
	www.math.fau.edu /Richman/Crypto/ass-cryp.htm (1230 words)

	Euler's theorem (Site not responding. Last check: 2007-11-03)
		is a Fermat pseudoprime to many different bases then
		(see Example 1.4.7 above), so by Euler's theorem,
		We can check this by hand without a computer: by the binomial expansion
	web.usna.navy.mil /~wdj/book/node42.html (60 words)

	Summer Session II of 2006 - MATH 173B (Introduction to Cryptology) (Site not responding. Last check: 2007-11-03)
		Lecture XIII Summary: primality test / PRIMES / FACTORIZE / Bertrand's postulate / pseudoprime / primality test (using pseudoprime) / Carmichael number
		Lecture XIV Summary: Euler pseudoprime / Solovay-Strassen primality test / strong pseudoprime / Miller-Rabin primality test
		If you are ever curious about GRH, click
	math.uci.edu /~tchoi/math173b (1793 words)

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