
	Where results make sense

Topic: Wieferich prime

Related Topics

prime minister

Prime Minister of the United Kingdom

Prime Minister of France

prime number

Prime Minister of Denmark

Prime Minister of Sweden

Prime Minister of Australia

Prime Minister of Italy

Prime Minister of Japan

Prime Minister of India

Prime Minister of Israel

List of Prime Ministers of Fiji

In the News (Mon 16 Nov 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	PlanetMath: Wieferich prime
		Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem.
		Prime numbers of this kind have been called Mirimanoff primes on occasion, but the name has not entered general mathematical use.
		This is version 4 of Wieferich prime, born on 2003-08-11, modified 2006-02-17.
	planetmath.org /encyclopedia/WieferichPrime.html (341 words)

	Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality - Numericana
		Wieferich primes are precisely the primes whose squares are Poulet numbers.
		Since p is prime, each of the (p+1) terms of the square bracket is congruent to 1 modulo p, and the whole sum is congruent to 1 modulo p.
		Wieferich primes are named after the German number theorist Arthur Wieferich (1884-1954) who established, in 1909, that any odd prime exponent in a counterexample to Fermat's Last Theorem would have to be such a prime.
	home.att.net /~numericana/answer/pseudo.htm (3003 words)

	Wieferich prime - Wikipedia, the free encyclopedia
		In mathematics, a Wieferich prime is prime number p such that p² divides 2
		The only known Wieferich primes are 1093 and 3511 (sequence A001220 in OEIS), found by W.
		From the definition of a Wieferich prime w it is, that 2
	en.wikipedia.org /wiki/Wieferich_prime (502 words)

	Puzzle 225. VSP (Very Sparse Primes)
		Evidently that all these small sets of "only known" primes, for a given property, are an invitation to discover the next one or the demonstration of the inexistence of more solutions.
		The only prime which is the difference of two prime squares.
		primes p1 that satisfies the congruence p2.p3-1 mod p1 =0 (p1, p2 and p3 are consecutive primes)
	www.primepuzzles.net /puzzles/puzz_225.htm (656 words)

	Arthur Josef Alwin Wieferich (Site not responding. Last check: 2007-11-05)
		There are 4 important papers by Wieferich, two in 1908 and two in 1909, when he was 24 and 25 years respectively, at that time a student of the University of Münster:
		Arthur Wieferich was born on 27th April 1884 in Münster, the son of a businessman.
		Wieferich as one of the gratifying recent advances in elementary number theory.
	www.numbertheory.org /obituaries/OTHERS/wieferich.html (415 words)

	Types of Prime Numbers
		- 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,...
		+ 1 is prime for n = 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154...
		The only known Wieferich primes are 1093 and 3511, which were both discovered in the early 20th century.
	people.bath.ac.uk /dm229/types.html (384 words)

	Wieferich primes and their relation with the squarefreness of F
		Wieferich primes and their relation with the squarefreness of F
		The author states that: ß(p^n) = p^(n-1)·ß(p) (for any prime p) This is not true if p = 1093 & n = 2: ß(p^n) == 364 p^(n-1)·ß(p) == 1093364 I believe that in general, the postulate in section 4.2 is not true for any Wieferich* prime.
		Of course, since no other Wieferich prime is known, and nobody knows how to find one, we can say that "There is no margin for Mersenne numbers to be anything else but squarefree." But unless we find the way to integrate Wieferich primes in an analyzable scheme, the proof is, as Jack Brennen claims, flawed.
	www.dybot.com /numbers/wiefe01.htm (1221 words)

	Active Distributed Computing Projects - Mathematics
		The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5.
		A Mersenne prime is a prime of the form 2P-1.
		A Proth prime is a prime number of the form k.2 n+1 where 2n > k.
	www.angelfire.com /theforce/distributedcomputing/math.html (1516 words)

	Trexle - Prime Numbers (Site not responding. Last check: 2007-11-05)
		Prime Numbers and Factoring - This page is a collection of links related to prime numbers and factoring of very large numbers.
		Formulae for Primes - The formula of Jones, Sato, Wada and Wiens: the set of primes is the set of positive values taken by this expression.
		Primes of the Form K * 2^N -1 - Tables for K=300 to 999.
	www.trexle.com /Directory/Top/Science/Math/Number_Theory/Prime_Numbers (1576 words)

	Sun Mountain Lodge
		The Lodge is the official residence of the Prime Minister of Australia in the national capital Canberra.
		The first Prime Minister to live in the lodge was Stanley Bruce, who moved in a few days before 9 May 1927, when the Old Parliament House was officially opened and the national capital changed from Melbourne to Canberra.
		This is a residence maintained for the official use of Prime Ministers when they need to perform official duties and extend official hospitality when in Sydney, but is not one that was ever intended to be the Prime Minister's (or anybody's) primary place of residence.
	www.artistbooking.com /trips/194/sun-mountain-lodge.html (847 words)

	Science - Math - Number Theory - Prime Numbers - Newsletter - News - Reviews - Education - Ratings
		Introductory prime number theory resources This excellent and thorough article is intended as a commentary to supplement the first half of a popular talk on the Riemann Hypothesis given by Peter Sarnak at a 1998 MSRI conference [a video recording is available here how the...
		Prime numbers Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
		Primes of the form K 2N -1 for K=301 to 399 Therefore the partial list of primes for N Nx+1, Nx+2 may or may not be complete up to the highest prime known.
	www.banner-net.com /Science/Math/Number_Theory/Prime_Numbers (2546 words)

	sci.crypt: every number has its own significance.....
		concatenated with the reverse of n is prime.
		641 is the smallest prime factor of 225+1.
		1031 is the length of the largest repunit that is known to be prime.
	www.derkeiler.com /Newsgroups/sci.crypt/2005-11/0545.html (11582 words)

	NSDL Metadata Record -- Wieferich Prime -- from MathWorld
		A Wieferich prime is a prime p which is a solution to the congruence equation 2^{p-1}\equiv 1\ \left({{\rm mod\ } {p^2}}\right).
		The first few Wieferich primes are 1093, 3511,...
		Wieferich, A. "Zum letzten Fermat'schen Theorem." J. reine angew.
	nsdl.org /mr/708599 (256 words)

	Proof that all prime-squared Mersenne divisors are Wieferich
		It has often been asked if all Mersenne numbers (with prime exponents) are square-free.
		Recall that the Wieferich primes are the primes p for which 2
		The only Wieferich primes less than 4,000,000,000,000 are 1093 and 3511.
	primes.utm.edu /notes/proofs/SquareMerDiv.html (203 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		is the largest number with the property that all smaller numbers relatively prime to it are prime.
		is the sum of the primes between its smallest and largest prime factor.
		is the smallest prime which is the sum of the cubes of the first few primes.
	home.comcast.net /~tdk500/Specialno.html (6437 words)

	Opera Directory (Site not responding. Last check: 2007-11-05)
		An introductory page on the theory of prime numbers with proofs of some important theorems including the infinitude of primes, Euclid's algorithm and the fundamental theorem of arithmetic.
		PRIMES is in P. A paper by Manindra Agrawal, Neeraj Kayal, Nitin Saxena that presents a polynomal-time algorithm that determines whether an input number n is prime or composite [PS].
		Since ancient times, mathematicians have been fascinated by problems concerning prime numbers, and many people have worked on the problem of determining ways to test if numbers are prime.
	portal.opera.com /web/?cat=48499 (1094 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		For any fixed prime p, the congruence k^p = k mod p^2 holds for p congruence classes modulo p^2: k=0 is the only solution among multiples of p, and the others are the (p-1)-st roots of unity in (Z/(p^2))^*; equivalently, they are the p-th powers in the ring Z/(p^2).
		Despite the irregularities noted two paragraphs earlier, it is appropriate in regions FAR from the horizontal axis ("far" meaning "relative to p") to think of the solution set as being the outcome of a random selection of points in the various vertical lines, with each point selected with probability 1/p.
		In a similar way, the Prime Number Theorem may be interpreted as stating that the prime numbers are roughly distributed as though selected by a random process with probability 1/log(p).
	www.math.niu.edu /~rusin/known-math/98/1093 (1238 words)

	[No title]
		The previous record was 10^9999+33603 (10000 digits), proved prime using ECPP on August 19, 2003.
		Wieferich primes are primes such that 2^(p-1) = 1 mod p^2.
		Sophie Germain primes P are such that P and 2P+1 are prime.
	www.loria.fr /~zimmerma/records/primes.html (648 words)

	Sun Hat
		A prime ''p'' > 5 is called a Wall-Sun-Sun prime if ''p''² divides :''F''(''p'' − (''p''5)), where ''F''(''n'') is the ''n''-th Fibonacci number and (''a''''b'') is the Legendre symbol of ''a'' and ''b''.
		Wall-Sun-Sun primes are named after D. Wall, Zhi Hong Sun and Zhi Wei Sun; Z. Sun and Z. Sun showed in 1992 that if the first case of Fermat's last theorem was false for a certain prime ''p'', then ''p'' would have to be a Wall-Sun-Sun prime.
		A Hi-hat or hihat is a type of cymbal and stand used as a typical part of a drum kit by percussionists in jazz, rock and roll, and other forms of contemporary popular music.
	aardogs.com /pages10/87/sun-hat.html (1032 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		prime q as factor of x^p + y^p, thus q^w
		- being a wieferich prime of needed "excess" of p-1
		primes at all in the integer-range which is accessible
	mathforum.org /kb/plaintext.jspa?messageID=4651136 (408 words)

	What's Special About This Number?
		is the smallest prime with 3 distinct digits that remains prime if one of its digits is removed.
		is the smallest multi-digit number so that the sum of the squares of its prime factors equals the sum of the squares of its digits.
		is the smallest prime where inserting the same digit between every pair of digits never yields another prime.
	www.stetson.edu /~efriedma/numbers.html (7473 words)

	In One Search - Start your search here (Site not responding. Last check: 2007-11-05)
		This page is a collection of links related to prime numbers and factoring of very large numbers.
		An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.
		For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7,11 and 13.
	www.inonesearch.com /directory.php?cid=48499 (850 words)

	Untitled Document
		prime p which is a solution to the
		Fermat's last theorem is false for exponent p, then p must be a Wieferich prime (Wieferich 1909).
		Silverman, J. "Wieferich's Criterion and the abc Conjecture." J.
	users.skynet.be /fa956617/math/topics/WieferichPrime.html (198 words)

	The Prime Glossary: Wieferich prime
		In 1909 Wieferich proved that if the first case of Fermat’s last theorem is false for the exponent p, then p satisfies this criterion.
		Since 1093 and 3511 are the only known such primes (and they have been checked to at least 32,000,000,000,000), this is a strong statement!
		But this is a long way from showing there are finitely many Wieferich primes.
	primes.utm.edu /glossary/page.php?sort=WieferichPrime (156 words)

	Internet-based Distributed Computing Projects - Active Projects - Mathematics
		The group is hosted by Team Prime Rib, but is open to everyone.
		There is a $500 (US) prize for the people who discover a new pair of Double Wieferich prime numbers.
		A Proth prime is a prime number of the form
	distributedcomputing.info /distrib-2002/ap-math.html (2710 words)

	Missouri Researchers Find World's Largest Known Prime Number - Sean Hannity Discussion (Site not responding. Last check: 2007-11-05)
		A prime number is a positive number divisible by only itself and 1 — 2, 3, 5, 7 and so on.
		Mersenne primes are a special category expressed as 2 to the "p" power minus 1, in which "p" also is a prime number.
		After the 23rd Mersenne prime was found at the University of Illinois, the mathematics department was so proud that the chair of their department, Dr. Bateman, had their postage meter changed to stamp "211213-1 is prime" on each envelope.
	www.hannity.com /forum/showthread.php?t=48092 (2487 words)

	Ivars Peterson's MathTrek - Conquering Catalan's Conjecture
		In 2000, Mihailescu proved that if additional solutions to the equation exist, the pair of exponents must be of a rare type known as double Wieferich primes.
		Peterson, I. Prime effort: Powerful conjecture may be proved.
		A definition of a double Wieferich prime pair can be found at http://mathworld.wolfram.com/DoubleWieferichPrimePair.html.
	www.maa.org /mathland/mathtrek_06_24_02.html (659 words)

	Mersenne Primes: History, Theorems and Lists
		It is not known if these last prime prime below is the 37th Mersenne prime as the region between them and the previous primes has not been completely tested.
		The 25th and 26th Mersenne primes were found by high-school students Laura Nickel and Landon Curt Noll, who, though they had little understanding of the mathematics involved, used Lucas' simple test on the local university's mainframe (CSUH's CDC 174) to find the next two primes.
		Slowinski's search for record primes is "not so organized as you would suppose" (his words), as he does not search systematically.
	w3.impa.br /~gugu/mersenne/mersenn1.txt (1687 words)

	Citations: Mathematics of Computation - distribution, condition (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		If n is prime, it is possible to use Rader s algorithm [126] which converts the transform into a circular convolution of size n #.
		The oscillations in the density of R primes leads me to conjecture that #(n) crosses Li(n) near 10 27 (Section 8.4) The remainder of the chapter analyzes the recurrence and discusses its consequences.
		Richard Crandall, Richard McIntosh, Carl Pomerance and other people found no other Wieferich prime up to 4 Delta 10 12 [2] The Maple program to compute F p (q max) is the following: F : proc(p,qmax) local q; for q from 3 by 2 to qmax do if Power(p,q 1) mod (q2) 1 then....
	citeseer.ist.psu.edu /context/784238/0 (1125 words)

Try your search on: Qwika (all wikis)