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Topic: Fermats little theorem

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	Fermat's little theorem - Wikipedia, the free encyclopedia
		Fermat's little theorem is the basis for the Fermat primality test.
		In this form, the theorem is used to justify the RSA public key encryption method.
		Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have
	en.wikipedia.org /wiki/Fermats_little_theorem (840 words)

	Number theory - Wikipedia
		Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity.
		The prime number theorem and the related Riemann hypothesis are examples.
		Warings problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well.
	nostalgia.wikipedia.org /wiki/Number_theory (545 words)

	A little known Theorem (Site not responding. Last check: 2007-11-07)
		Fermat’s Little Theorem was created after his so called Last Theorem, sometime around the 1640’s.
		It is used more widely than his Last Theorem in modern mathematics, as it is of great use to students of the subject.
		Fermat's Little Theorem can be used in number theory and for testing large primes.
	students.bath.ac.uk /ma3pab/little.html (165 words)

	Fermats last theorem - ExampleProblems.com
		Fermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem) is one of the most famous theorems in the history of mathematics.
		While the theorem itself has no known direct use (i.e., it has not been used to prove any other theorem), it has been shown to be connected to many other topics in mathematics, and is not merely an unimportant mathematical curiosity.
		Fermat did not publish proofs for the vast majority of his theorems, including those theorems for which mathematical historians believe he actually had a proof.
	www.exampleproblems.com /wiki/index.php/Fermats_last_theorem (1644 words)

	Pseudoprime: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-07)
		Fermats little theorem states that if p is a prime number, then for any integer a,...
		A number x that is a pseudoprime for all values of a that are coprime to x is called a Carmichael number[Follow this hyperlink for a summary of this subject].
		Fermats last theorem (sometimes abbreviated as flt and also called fermats great theorem) is one of the most famous theorems in the history of mathem...
	www.absoluteastronomy.com /encyclopedia/p/ps/pseudoprime.htm (1349 words)

	Full Proof of Fermat's Last Theorem (Site not responding. Last check: 2007-11-07)
		I think its most likely that Fermat's theorem is either a cruel joke Fermat played on all future mathematics, or the proof he envisioned when he wrote the note was incorrect or incomplete.
		It's not the theorem itself that has imprisoned many a mathmatitian in their studies, it's his 'truly marvelous proof' Had he simply noted the theorem itself in the margins without this last statement i doubt history would have made much of it.
		Not so with the little theorem which was almost certainly proved by fermat but which at any rate was proven by many in the 1700's and even prior.
	digg.com /science/Full_Proof_of_Fermat_s_Last_Theorem (872 words)

	Pseudoprime
		In general, an integer which has a certain property shared by all prime numbers, but is itself not prime, is called a pseudoprime for that particular property.
		The most important class of pseudoprimes come from the Fermat's little theorem and hence they are called Fermat pseudoprimes.
		If the user does not require the test to be completely accurate (say, he might tolerate a very small chance that a composite number is declared to be prime), there are fast algorithms based on Fermat's Little Theorem.
	www.ebroadcast.com.au /lookup/encyclopedia/ps/Pseudoprime.html (422 words)

	Prime number - Wikipedia
		An important result is the fundamental theorem of arithmetic, which states that every natural number can be written as a product of primes, and in essentially only one way.
		An efficient way to compute a list of all the prime numbers up to a given limit is the algorithm called the "Sieve of Eratosthenes".
		For a random large number (say, up to a few thousand digits), you can test for primality with Fermats little theorem or the Miller-Rabin test.
	nostalgia.wikipedia.org /wiki/Prime_number (788 words)

	Fermat's little theorem
		145 you will find: 'Fermat states a result of which an important theorem, now known as the "little Fermat theorem," is a consequence.' This indicates to me that someone had coined the term "little Fermat theorem" by the time Uspensky and Heaslet published their book in 1939.
		So this little bit of detective work reveals that the term "little Fermat theorem" probably first appeared sometime between 1936 and 1939 and was in common usage by 1945.
		Translation: "There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it." So at least in German, this seems to have been a term in common use, in 1913.
	www.spd.dcu.ie /johnbcos/fermat's_little_theorem.htm (1565 words)

	Search Results for theorem
		The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years.
		It is nevertheless certain that the theorem on the sum of the three angles of the triangle should be considered one of those fundamental truths that are impossible to contest and that are an enduring example of mathematical certitude.
		Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=theorem&CONTEXT=1 (15920 words)

	Fermat's last theorem : Fermats Last Theorem (Site not responding. Last check: 2007-11-07)
		The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of famous Diophantus' Arithmetica, "I have discovered a truly remarkable proof but this margin is too small to contain it".
		The reason why this statement is so significant is that all the other theorems proposed by Fermat were settled either by proofs he supplied, or by more rigorous proofs supplied afterwards.
		The methods used by Wiles were unknown when Fermat was writing, and it seems inconceivable that Fermat managed to derive all the necessary mathematics to demonstrate the same solution (in the words of Andrew Wiles, "it's impossible; this is a 20th century proof").
	www.findword.org /fe/fermats-last-theorem.html (884 words)

	Number Theory
		This comment was found in Fermat’s copy of the ‘Arithmetica’ next to one of Diophantus’s theorems that commented on expressing a perfect square as the sum of two squares.
		All that this statement lacks is Fermat’s proof, which he sadly did not demonstrate before his death in 1665.
		In mathematical terms, the theorem was as important a proof to mathematics as, say, completing the Genome Project and cataloguing every gene, or placing man on the moon.
	people.bath.ac.uk /mta20/page4.html (434 words)

	Fermat's Last Theorem: Fermat's Little Theorem
		Despite its name, Fermat's Little Theorem is one of Pierre de Fermat's most important theorems.
		Fermat first presented it without proof in one of his letters in 1640.
		This theorem is also the foundation of Sophie's Proof (see here for the details), an important result from the mathematician Sophie Germain.
	fermatslasttheorem.blogspot.com /2005/08/fermats-little-theorem.html (355 words)

	Simple solution of FLT? Text - Physics Forums Library
		Fermat was a genius but like lots of us he got many things wrong you'd be surprised how many old well known results attributed to the greats were never properly proven by them.
		N =4 was presented by Fermat himself, one of the few things he did prove, and by the method of infinite descent.
		07-09-2005, 12:09 PM Since Fermat published his proofs of special cases AFTER he wrote that he had a way of proving it for all n, that is almost certain.
	www.physicsforums.com /archive/index.php/t-81230.html (1646 words)

	Cryptography-Digest Digest #957
		Current Sources use Fermat tests to the bases of 2, 3, 5, 7, 11, 13 and 17 as well as make several other strong checks for pseudoprimes (composite numbers that pass a Fermat test to the base 2).
		The density of actual 1024 bit prime numbers is about 1.41e-3; the chances of getting a pseudoprime from a number of 1024 bits picked at random instead of a real prime are at least 2.13e41 to 1, and are thought to be closer to 3.2e85 to 1.
		Fermats little theorem says that a^p mod p = p always if p is prime, or a^(p-1) = 1 mod p.
	www.mail-archive.com /cryptography-digest@senator-bedfellow.mit.edu/msg02155.html (3263 words)

	Tori Spelling (Site not responding. Last check: 2007-11-07)
		'Under The Pink' is the definate mid point in vision between the structured, catchy honest confession of 'Little Earthquakes' and the experimental, often formless cryptic 'Boys For Pele.' I feel that with her debut Tori didn't really show off her lyrical...
		Strange Little Girls is a satisfying addition to Tori Amos' body of work.
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	www.freeglossary.com /Tori_Spelling (489 words)

	Number Theory (Site not responding. Last check: 2007-11-07)
		Of all the theorems that were outlined by Fermat, one of the most important theorems he stated was in a letter of 1640.
		Many have tried to find a pattern to these so-called 'perfect pairs' of numbers, and it was only until 1750, when Euler found a way of generating them, that it was resolved.
		However, with reference to Fermats little theorem, his statement was;
	people.bath.ac.uk /mta20/page5.html (201 words)

	How to find primes and prove primality (merged version)
		Theorem 2: Suppose n-1 = FR, where F>R, gcd(F,R) is one and the factorization of F is known.
		It is because there is a theorem similar to Fermat's Little theorem that we can use here--but first we must do a little ground work.
		Theorem: Suppose that a and p are relatively prime integers with p > 1.
	primes.utm.edu /prove/merged.html (6842 words)

	Lecture 5
		Proving the theorem was hard because we do not precisely know how the primes are distributed.
		The simple Fermat test is too unreliable for certain numbers, particularly, we cannot prove anything of the kind that with probability at least alpha > 0 a non-prime will be unveiled.
		Theorem: For an arbitrary odd number n > 4, if n is a prime, then the improved Fremat test returns true for all 1 < a < n, else it return true for less than one quarter of the a.
	www.cs.umu.se /kurser/TDBC91/VT02/lec5.html (2121 words)

	MathLinks Math Forum :: View topic - Fermat's Theorem
		I think that the actual theorem is somewhat different, and the above formula is just like an offshoot of it or something.
		The original theorem (Euler's phi theorem) is used (or at least I use) very often for problems asking to find the last x digits of an enormous number.
		Fermats little theorem tells you instantly that it will repeat..
	www.mathlinks.ro /Forum/post-157135.html (1543 words)

	History Of Perfect Numbers (Site not responding. Last check: 2007-11-07)
		I don't doubt that Frenicle de Bessy got there earlier, but I have only begun and without doubt these propositions will pass as very lovely in the minds of those who have not become sufficiently hypocritical of these matters, and I would be very happy to have the opinion of M Roberval.
		Fermat found this theorem as a consequence of his investigation into Perfect Numbers.
		Fermats results on Perfect Numbers interested marsenne greatly and led to him producing a claim of his own which fascinated mathematicians for many years.
	students.bath.ac.uk /ma2le/History.html (1527 words)

	Proofs of Fermat's little theorem - All About All (Site not responding. Last check: 2007-11-07)
		Our use of this cancellation law in the above proof of Fermat's little theorem was valid, because the numbers 1, 2,..., p − 1 are certainly not divisible by p (indeed they are smaller than p).
		By Lagrange's theorem, k divides the order of G, which is p − 1, so p − 1 = km for some positive integer m.
		This proof uses induction to prove the theorem for all integers a ≥ 0.
	www.answers-zone.com /article/Proofs_of_Fermat%27s_little_theorem (2165 words)

	History of Prime Numbers (Site not responding. Last check: 2007-11-07)
		Unfortunatly this algorithm is very slow, it has exponential complexity, meaning it runs exponentially in the number of digits in the number n.
		Fermats little theorem, which tells you, that if p is a prime and a is a number not divisble by p, then a
		This theorem resulted in a number of probalistic algorithms, which is the most common way of finding primes today.
	www.daimi.au.dk /~omega/Primes/history.html (335 words)

	[No title]
		To every theorem stated in these mathematical studies, there corresponds a name of one of our contributors. Names of the theorems in these studies are chosen from the set of the
		The most Interesting thing about them is, they combine the irrational numbers with the numberstheory which is quite a highly interesting fact.(example ; there exists different forms of Fermats little theorem with irrational and imaginary numbers.
		Theorem of EL'Muhsi (Geometrical interpretation and generation of
	www.geocities.com /timeparadox (609 words)

	Fermat little theorem ... problem (Site not responding. Last check: 2007-11-07)
		According to fermats little theorem, a number n is prime if bor any number b such that gcd(b,n) = 1 one has: b^(n-1) = 1 mod n is this right?
		Monday, 10 March 2003 11:35:47 AM the theorem is sn 'if' not an 'only if theorem.
		Monday, 10 March 2003 11:35:51 AM the theorem is sn 'if' not an 'only if theorem.
	alex.edfac.usyd.edu.au /chatrooms/Maths/304862368.html (412 words)

	CALCULUS OF VARIATIONS - LoveToKnow Article on CALCULUS OF VARIATIONS (Site not responding. Last check: 2007-11-07)
		The contributions of the Greek geometry to the subject consist of a few theorems discovered by one Zenodorus, of whom little Earl is known.
		A simple example is furnished by the problem of forming the equations of the path of a ray of light in a variable medium.
		Path of According to Fermats principle, the integral fuds is a a ray.
	www.1911ency.org /V/VA/VARIATIONS_CALCULUS_OF.htm (5115 words)

	Finding Primes! (Site not responding. Last check: 2007-11-07)
		One test that is used though is Fermats Little Theorem.
		Using this theorem you can test whether a number is composite.
		When the numbers you want to test for primality increase, the tests for primality become more complicated and more time consuming.
	www.bath.ac.uk /~ma2leb/primef.html (233 words)

	Syllabus for MATH 4161 (Section 001) (Site not responding. Last check: 2007-11-07)
		MATH 3163 with a grade of C or better or consent of the department.
		Fermats Little Theorem and Euler's generalization (Section 5.3, Chapter 7)
		Homework will be handed in nearly every week, and will be usually due on Monday.
	www.math.uncc.edu /~ghetyei/courses/old/S02.4161/syllabus.html (116 words)

	Fermat's little theorem
		If P is a prime number then for any number a, (a
		This theorem is useful for testing if a number is not prime, though it can't tell if a number is prime.
		As was his habit, Pierre de Fermat didn't provide a proof (this time saying "I would send you the demonstration, if I did not fear its being too long").
	www.daviddarling.info /encyclopedia/F/Fermats_little_theorem.html (166 words)

	Prime Numbers - Page 2
		You have to understand that a Mersenne prime is a special case where mathematicians and programmers use special criterion to work with.
		This particular property does not quite characterize primes, since some non prime numbers also obey the conclusion of fermats little theorem, but a small enhancement of it does so.
		Note that theorem 5.1 in either the original or revised paper suggest an upper bound of
	www.physicsforums.com /showthread.php?t=65118&page=2 (1948 words)

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