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Topic: Sierpinski number

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	PlanetMath: Sierpinski number
		That such numbers exist is amazing, and even more surprising is that there are infinitely many of them (in fact, infinitely many odd ones).
		The smallest known Sierpinski number is 78557, but it is not known whether or not this is the smallest one.
		This is version 4 of Sierpinski number, born on 2003-09-01, modified 2007-08-22.
	planetmath.org /encyclopedia/SierpinskiNumbers.html (158 words)

	Wacław Sierpiński - Wikipedia, the free encyclopedia
		Sierpinski began to study set theory and, in 1909, he gave the first ever lecture course devoted entirely to the subject.
		Sierpinski was awarded the scientific prize of the first degree in 1949.
		Waclaw Sierpinski is interred in the Powązki Cemetery, Warsaw, Poland.
	en.wikipedia.org /wiki/Waclaw_Sierpinski (799 words)

	The Prime Glossary: Sierpinski number
		In 1960 Sierpinski showed that there were infinitely many such numbers k (all solutions to a family of congruences), but he did not explicitly give a numerical example.
		The congruences provided a sufficient, but not necessary, condition for an integer to be a Sierpinski number.
		It is conjectured that 78557 is the smallest Sierpinski number becase for most of the smaller numbers we can easily find a prime (in fact, for about 2/3rds of the numbers k there is a prime with n less than 9).
	primes.utm.edu /glossary/page.php?sort=SierpinskiNumber (470 words)

	What's Special About This Number? (Site not responding. Last check: )
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7257 words)

	Sierpinski number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: )
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a Sierpinski number is an odd (The number 1 and any other number obtained by adding 1 to it repeatedly) natural number k such that integers of the form k2
		In 1960 (Click link for more info and facts about Waclaw Sierpinski) Waclaw Sierpinski proved that there are (Click link for more info and facts about infinite) infinitely many odd (Any of the natural numbers (positive or negative) or zero) integers that when used as k produce no primes.
		As of 2004, all but ten of these numbers had been shown to produce primes, and were thus eliminated as possible Sierpinski numbers.
	www.absoluteastronomy.com /encyclopedia/s/si/sierpinski_number.htm (322 words)

	Sierpinski number - Wikipedia, the free encyclopedia
		In mathematics, a Sierpinski number is an odd natural number k such that integers of the form k2
		To show that 78,557 really is the smallest Sierpinski number, one must show that all the odd numbers smaller than 78,557 are not Sierpinski numbers.
		As of 2005, all but eight of these numbers had been shown to produce primes, and were thus eliminated as possible Sierpinski numbers.
	en.wikipedia.org /wiki/Sierpinski_number (232 words)

	Manpage of XLOCK (Site not responding. Last check: )
		In braid mode it is the upper bound number of strands.
		In sballs mode it is the number of spheres.
		In triangular mode it is the number of mountains.
	linux.math.tifr.res.in /manuals/man/xlock.html (5420 words)

	Sierpinski biography
		Sierpinski was lucky for the lector changed the mark on his Russian language course to 'good' so that he could take his degree.
		Sierpinski graduated in 1904 and worked for a while as a school teacher of mathematics and physics in a girls school in Warsaw.
		In 1916, during his time in Moscow, Sierpinski gave the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Sierpinski.html (1798 words)

	Sierpinski number (Site not responding. Last check: )
		In mathematics, a Sierpinski number is a positive, odd, number k such that integers of the form k2
		Seventeen or Bust, a distributed computing project, is trying all untested numbers that are less than 78,557 to see whether they are Sierpinski numbers.
		When the project started there were 17 numbers k for which such a prime was unknown (hence the name of the project) and in the first year of its existence the project succeeded in finding 6 more primes; hence (as of April 2004) there are 11 more ks to be tested.
	www.sciencedaily.com /encyclopedia/sierpinski_number (291 words)

	All You Ever Wanted to Know About Pascal's Triangle and more
		The sum of the numbers in the consecutive rows shown in the diagram are the first numbers of the Fibonnacci Sequence.
		The second number is equal to the number of vertexes of the polygon.
		When all the odd numbers (numbers not divisible by 2) in Pascal's Triangle are filled in (fl) and the rest (the evens) are left blank (white), the recursive Sierpinski Triangle fractal is revealed (see figure at near right), showing yet another pattern in Pascal's Triangle.
	ptri1.tripod.com (1016 words)

	Neglected Gaussians
		The numbers 220 and 284 are amicable, because each is the sum of the divisors of the other.
		Perfect numbers are length 1 cycles, Amicable numbers are length 2 cycles, Sociable numbers are length 3+ cycles.
		Sierpinski Number is a number k such that k 2
	www.mathpuzzle.com /Gaussians.html (723 words)

	Searching for Robinson primes using values of k with low Nash weight
		In fact the method may be modified specifically to find Sierpinski numbers, by ignoring k as soon as any value of n between 1 and 1000 survives the trial division, although in practice it is convenient to preserve the possibility of one or two exceptions.
		I am therefore confident that the number of Sierpinski numbers not identified in the range of k searched is very small and may, in fact, be zero.
		This idea stretches to all Sierpinski numbers, that is, for a given covering set and modulus, all Sierpinski numbers k such that k > 2P have an associated base value k' such that k' < 2P.
	www.glasgowg43.freeserve.co.uk /siernash.htm (3040 words)

	Sierpinski Gasket
		Instead of removing the central third of a triangle, the central square piece is removed from a square sliced into thirds horizontally and vertically.
		The construction of the 3 dimensional version of the gasket follows similar rules for the 2D case except that the building blocks are square based pyramids instead of triangles.
		The essence of the technique is that an N dimensional object is illustrated with a stack of N-1 dimensional objects.
	astronomy.swin.edu.au /~pbourke/fractals/gasket (1697 words)

	Math Games: Gaussian Numbers
		Similar searches for Gaussian Integers are unknown to me. For complex numbers, the Zetagrid project has collected the first 687 billion zeroes of the Riemann Zeta function.
		For almost every interesting number theory question, there is a similar question in the Gaussian integers.
		Sierpinski Number is a number k such that k•2
	www.maa.org /editorial/mathgames/mathgames_03_15_04.html (707 words)

	Sierpinski Gasket and Tower of Hanoi
		As was discovered by Ian Stewart, puz(Tower of Hanoi) has a surprising relationship to the Sierpinski gasket (also known as the Sierpinski triangle) and, therefore, to Pascal's triangle.
		In the Tower of Hanoi puzzle, disks stacked on one peg are to be moved to another with, perhaps an intermediate stop at a third, auxiliary peg.
		Every disk may be said to be at one of the pegs and, for a specific configuration, be associated with one of the numbers: 1, 2, 3.
	www.cut-the-knot.org /triangle/Hanoi.shtml (1112 words)

	Shane's Mathematics Page
		These values of k are called Sierpinksi Numbers due to the fact that the Polish mathematician Waclaw Sierpinski not only showed that they existed but, even more surprisingly, proved that there were an infinite amount of them.
		At the beginning of 2002 only 17 numbers less that 78557 were still in doubt (every other integer less than 78557 had been shown to produce a prime number for at least one value of n).
		Their project called Seventeen or Bust (www.seventeenorbust.com) has, in its first year, proved that 5 of the values produce prime numbers and so are not Sierpinski Numbers.
	www.geocities.com /shanessiteuk/mathematics.htm (1308 words)

	Distributed Computing - Active Projects - Mathematics
		The project looks for Proth prime numbers in which, for a number k, if every possible choice of n results in a composite (non-prime) Proth number N, k is a Sierpinski number.
		The number of composited was reduced to 2500 as of September 10, 2005.
		You can reserve numbers manually through the project website and factor them with your favorite factoring client application (GMP-ECM is reocmmended), or you can use the ECMclient application and automatically reserve numbers and submit results (use the ecmserver childers.myip.org, port 34).
	distributedcomputing.info /ap-math.html (5827 words)

	Sierpinski problem
		In 1962, John Selfridge discovered the Sierpinski number k = 78557, which is now believed to be in fact the smallest such number.
		The integer k = 78557 is the smallest Sierpinski number.
		= 7238 is the number of those k for which k.2 + 1 is a prime, the first one obviously being k = 1.
	www.prothsearch.net /sierp.html (588 words)

	The Prime Sierpinski Problem (Site not responding. Last check: )
		Prime numbers are numbers, which are divisible by 1 and by themselves and not by any other numbers.
		We are looking at all k’s below this number and trying to prove that they are not sierpinski numbers and thus studying the distribution of primes of the forum k*2^n+1.
		When a k is proved that it is not a sierpinski number the k is eliminated.
	www.geocities.com /eharsh82 (444 words)

	49. Sierpinski-like numbers
		In 1962, John Selfridge discovered the Sierpinski number k = 78557, which is now believed to be in fact the smallest such number.
		As in the Sierpinski and Riesel Problems, one could conjecture that 79 is the smallest such k.
		In this connection, the number P = 2.3.5.7.11.13.17.19.31.37.41.61.73.97.109.151.181.241.257.331.673 = 8098504028425981183736114374652670 is essential, because every number k = k1 + h.P (h = 0, 1, 2,...) is also a Sierpinski number, and every k = k2 + h.P is also a Riesel number (infinitely many).
	www.primepuzzles.net /problems/prob_049.htm (2402 words)

	17 Or Bust (Site not responding. Last check: )
		Numbers of this form are called Proth numbers.
		Most number theorists believe that this is the smallest, but it hasn't yet been proven.
		In order to prove it, we have to show that every single k less than 78,557 is not a Sierpinski number, and to do that, we have to find some n that makes k * 2^n + 1 prime.
	www.gameland.gr /sob.htm (507 words)

	Ivars Peterson's MathTrek - A Remarkable Dearth of Primes
		A multiplier k with this property is called a Sierpinski number.
		With five Sierpinski candidates cracked in rapid succession, there are only 12 more to go.
		Additional information about Sierpinski numbers can be found at http://primes.utm.edu/glossary/page.php?sort=SierpinskiNumber and http://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html.
	maa.org /mathland/mathtrek_01_13_03.html (557 words)

	The Sierpinski Tetrahedron
		The starting shape (initiator) for the Sierpinski tetrahedron is a tetrahedron, and it grows according to a rule (generator) whereby each tetrahedron is replaced in the next stage by four tetrahedra set tip-to-tip.
		The number of tetrahedra at any given stage, or iteration, is always the number of tetrahedra in the previous iteration, multiplied by four.
		Beginning with the stage-4 Sierpinski tetrahedron, there is hidden volume in the shape of tetrahedra contained in the interior of the Complement, exactly 24 for the stage-4.
	www.public.asu.edu /~starlite/sierpinskitetrahedron.html (1227 words)

	Sierpinski Triangle
		Without a doubt, Sierpinski's Triangle is at the same time one of the most interesting and one of the simplest fractal shapes in existence.
		The geometric construction of the Sierpinski triangle is the most intuitive way to generate this fascinating fractal; however, it is only the tip of the Sierpinski iceberg.
		For example, the Sierpinski Triangle is a canonical example of a shape known as a fractal.
	www.zeuscat.com /andrew/chaos/sierpinski.html (1278 words)

	Problem 29.- Brier Numbers
		Sierpinski number k = 78557, which is now believed to be in fact the smallest such number.
		This 34 digit number is probably the smallest possible one.
		Yves Gallot has wrote in his own site a note about his search for Brier numbers that maybe will be of interest to Brier numbers hunters.
	primepuzzles.net /problems/prob_029.htm (1091 words)

	The Magic Sierpinski Triangle
		This design is called Sierpinski's Triangle (or gasket), after the Polish mathematician Waclaw Sierpinski who described some of its interesting properties in 1916.
		The large blue triangle consists of three smaller blue triangles, each of which itself consists of three smaller blue triangles, each of which..., a process of subdivision which could, with adequate screen resolution, be seen to continue indefinitely.
		That's the fourth choice on the display panel, and in that case Serendip is using all three vertices an equal number of times but is picking them randomly, so that there is no pattern whatsoever in its choices.
	serendip.brynmawr.edu /playground/sierpinski.html (973 words)

	2CPU.com - The one stop source for everything SMP!
		DC Project Profile: Seventeen Or Bust A Sierpinski number is a value k such that k*(2^n + 1) is never prime for all values of n greater than or equal to 1.
		currently, the smallest known Sierpinski number is 78557 and according to the Sierpinski conjecture, it is the smallest such number.
		As of early 2002, all but 17 smaller values of k were proven not to be a Sierpinski number and thus the Seventeen or Bust project was born to determine whether any of the seventeen candidates were in fact a Sierpinski number and would become the smallest known number of that type.
	www.2cpu.com /story.php?id=3236 (733 words)

	Seventeen or Bust: Home
		The Sierpinski problem itself deals with numbers of the form N = k * 2^n + 1, for any odd k and n > 1.
		The Sierpinski problem itself is: "What is the smallest Sierpinski number?" (For a more rigorous mathematical discussion of the problem, see prothsearch.net's Sierpinski Problem page.)
		Most number theorists believe that this is the smallest, but it hasn't yet been proven.
	www.seventeenorbust.com (1212 words)

	Riesel and Sierpinski Numbers
		A Brier number is an odd number k that is simultaneously a Sierpinski number and a Riesel number.
		It is possible to generate a Sierpinski number from a specific covering pattern by generating a list of congruences of the form k
		It is appropriate to obtain firstly a set of Nash congruences for k as a Sierpinski number, convert these to Sierpinski congruences, and for each overlapping prime, consider the Sierpinski congruence as a Riesel congruence and produce the associated Nash congruence.
	www.glasgowg43.freeserve.co.uk /brier2.htm (906 words)

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