en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

2014-01-21T07:16:09Z

Further reading: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 09:16, 21 January 2014
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	~~The individual who wrote~~ the ~~post~~ is ~~called Jayson Hirano~~ and ~~he completely digs~~ that ~~title~~. ~~Office supervising is what she does~~ for ~~a residing~~. ~~Mississippi is where his home is~~. ~~To play domino is something I really appreciate performing~~.<br><br>~~My homepage~~; ~~online psychic reading~~ ([~~http~~://~~myoceancounty~~.~~net~~/~~groups~~/~~apply-these-guidelines~~-when~~-gardening-and-grow~~/ ~~mouse click~~ the ~~up coming website~~ page])		{{Cleanup\|date=October 2010}}
			[[File:Wheel factorization-n=30.svg\|thumb\|Wheel factorization with n=2x3x5=30. No primes will occur in the yellow areas.]]
			'''Wheel factorization''' is a graphical method for manually performing a preliminary to the [[Sieve of Eratosthenes]] that separates [[prime number]]s from [[Composite number\|composites]]. Start by writing the natural numbers around circles as shown below. Prime numbers in the innermost circle have their multiples in similar positions as themselves in the other circles, forming spokes of primes and their multiples. Multiples of the prime numbers in the innermost circle form spokes of composite numbers in the outer circles.

			==Procedure==
			# Find the first few prime numbers. They are known or can be found quickly using [[Sieve of Eratosthenes]].
			# Multiply the prime numbers together to give the result ''n''.
			# Write 1 to ''n'' in a circle. This will be the inner-most circle.
			# Taking ''x'' to be the number of circles written so far, continue to write ''xn'' + 1 to ''xn'' + ''n'' in another circle around the inner-most circle, such that ''xn'' + 1 is in the same position as (''x'' − 1)''n'' + 1.
			# Repeat step 4 until the largest number to be tested for primality.
			# Strike off the number 1.
			# Strike off the spokes of prime numbers (found in step 1) with its multiples without striking off the numbers in the inner-most circles.
			# Strike off the spokes of all multiples of prime numbers found in step 1.
			# The remaining numbers in the wheel contain mostly prime numbers. Use other methods such as Sieve of Eratosthenes to remove the remaining non-primes.

			==Example==
			[[File:Wheel factorization-n=6.svg\|thumb\|Wheel factorization with n=2x3=6]]
			1. Find the first 2 prime numbers—2 and 3.

			2. ''n'' = 2 · 3 = 6

			3.
			1 2 3 4 5 6

			4. ''x'' = 1.<br> ''xn'' + 1 = 1 · 6 + 1 = 7.<br> (''x'' + 1)''n'' = (1 + 1) · 6 = 12.<br> Write 7 to 12 with 7 aligned with 1.
			1 2 3 4 5 6
			7 8 9 10 11 12
			5. ''x'' = 2.<br> ''xn'' + 1 = 2 · 6 + 1 = 13.<br> (''x'' + 1)''n'' = (2 + 1) · 6 = 18.<br> Write 13 to 18.<br> Repeat for the next few lines.
			1 2 3 4 5 6
			7 8 9 10 11 12
			13 14 15 16 17 18
			19 20 21 22 23 24
			25 26 27 28 29 30
			6 and 7. Sieving
			<s>1</s> 2 3 4 5 6
			7 <s>8</s> <s>9</s> 10 11 12
			13 <s>14</s> <s>15</s> 16 17 18
			19 <s>20</s> <s>21</s> 22 23 24
			25 <s>26</s> <s>27</s> 28 29 30
			8. Sieving
			<s>1</s> 2 3 <s>4</s> 5 <s>6</s>
			7 <s>8</s> <s>9</s> <s>10</s> 11 <s>12</s>
			13 <s>14</s> <s>15</s> <s>16</s> 17 <s>18</s>
			19 <s>20</s> <s>21</s> <s>22</s> 23 <s>24</s>
			25 <s>26</s> <s>27</s> <s>28</s> 29 <s>30</s>
			9. Resultant list contains a non-prime 25. Use other methods of sieve to arrive at
			2 3 5 7 11 13 17 19 23 29

			==Analysis and Computer implementation==
			Given a number k > n, we know that it is not prime if k mod n and n are not relatively prime. The fraction of numbers that the wheel sieve eliminates is 1 - [[Euler's totient function\|phi]] (n) / n, which is also the efficiency of the sieve. From the properties of phi, it can easily be seen that the most efficient sieve smaller than x is the one where <math> n = p_1 p_2 ... p_i < x </math> and <math> n * p_{i+1} >= x </math>. It can also be demonstrated that this efficiency rises very slowly for large n.

			To be of maximum use on a computer, we want the numbers that are smaller than n and relatively prime to it as a set. Using a few observations, the set can easily be generated :
			# Start with <math> S_2 = \{1\} </math>, which is the set for <math> n=2 </math>.
			# Let <math> S_n + k </math> be the set where k has been added to each element of <math> S_n </math>.
			# Then <math>S_{n p_{i+1}} = F_{p_{i+1}} [S_n \cup S_n + n \cup S_n + 2n \cup ... \cup S_n + n (p_{i+1} - 1)]</math> where <math> F_x </math> represents the operation of removing all multiples of x.
			# 1 and <math> p_{i+1} </math> will be the two smallest of <math> S_n </math> when <math> n > 2 </math> removing the need to compute prime numbers separately
			# The set is symmetrical around <math> n / 2 </math>, reducing storage requirements.

			==External links==
			* [http://primes.utm.edu/glossary/page.php?sort=WheelFactorization Wheel Factorization]
			* [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.835 Improved Incremental Prime Number Sieves] by Paul Pritchard

			{{number theoretic algorithms}}

			{{DEFAULTSORT:Wheel Factorization}}
			[[Category:Primality tests]]

en>Palosirkka: license

2012-01-27T09:54:39Z

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The individual who wrote the post is called Jayson Hirano and he completely digs that title. Office supervising is what she does for a residing. Mississippi is where his home is. To play domino is something I really appreciate performing.<br><br>My homepage; online psychic reading ([http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ mouse click the up coming website page])

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en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

en>Palosirkka: license