Quasi-Newton method - Revision history

en>Velvel2: /* Further reading */ dead link

2014-12-11T05:41:17Z

131.111.243.142: /* Description of the method */

2014-02-22T01:03:27Z

Description of the method

← Older revision		Revision as of 03:03, 22 February 2014
Line 1:		Line 1:
	~~An '''Achilles number'''~~ is ~~a number that is [[Powerful number\|powerful]] but not a [[perfect power]].<ref~~ name~~=mw>{{MathWorld\|urlname=AchillesNumber\|title=Achilles Number}}~~		The author is known by the name of Numbers Lint. One of the extremely very best issues in the globe for me is to do aerobics and now I'm attempting to earn cash with it. Managing individuals is his profession. California is exactly where her house is but she needs to move simply because of her family.<br><br>Also visit my web page - [http://torontocartridge.com/uncategorized/the-ideal-way-to-battle-a-yeast-infection/ home std test]
	~~</ref> A positive integer ''n'' is a powerful number if, for every [[prime factor]] ''p''~~ of ~~''n'', ''p''<sup>2</sup> is also a divisor. In other words, every prime factor appears at least squared in the factorization~~. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as ''m<sup>k</sup>'', where ''m'' and ''k'' are positive integers greater than 1.

	~~Achilles numbers are named after [[Achilles]], a hero~~ of the ~~[[Trojan war]], who was also powerful but imperfect.~~

	~~==Sequence of Achilles numbers==~~
	A number ''n'' = ''p''<sub>1</sub><sup>''a''<sub>1</sub></sup>''p''<sub>2</sub><sup>''a''<sub>2</sub></sup>…''p''<sub>''k''</sub><sup>''a''<sub>''k''</sub></sup> is [[powerful number\|powerful]] if min(''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''k''</sub>) ≥ 2. If in ~~addition gcd(''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''k''</sub>) = 1~~ the ~~number~~ is ~~an Achilles number.~~

	~~The Achilles numbers up~~ to ~~5000 are:~~
	:72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 {{OEIS\|id=A052486}}.

	~~The smallest pair of consecutive Achilles numbers is:<ref>Carlos Rivera, ''The Prime Puzzles~~ and ~~Problem Connection~~'~~', [http://www~~.~~primepuzzles~~.~~net/problems/prob_053.htm Problem 53]</ref>~~

	~~: 5425069447 = 7<sup>3</sup> × 41<sup>2</sup> × 97<sup>2</sup>~~
	~~: 5425069448 = 2<sup>3</sup> × 26041<sup>2</sup>~~

	~~==Examples==~~
	~~108~~ is ~~a powerful number. Its [[prime factorization]]~~ is ~~2<sup>2</sup> · 3<sup>3</sup>, and thus its prime factors are 2 and 3~~. ~~Both 2~~<~~sup~~>2<~~/sup~~> ~~= 4 and 3<sup>2<~~/~~sup> = 9 are divisors of 108. However, 108 cannot be represented as ''m<sup>k<~~/~~sup>'', where ''m'' and ''k'' are positive integers greater than 1, so 108 is an Achilles number.~~

	~~Finally, 784 is not an Achilles number~~. ~~It is a powerful number, because not only are 2 and 7 its only prime factors, but also 2<sup>2<~~/~~sup> = 4 and 7<sup>2<~~/~~sup> = 49 are divisors of it. Nonetheless, it is~~ a ~~perfect power:~~

	~~:<math>784=2^4 \cdot 7^2 = (2^2)^2 \cdot 7^2 = (2^2 \cdot 7)^2 = 28^2. \, <~~/~~math>~~

	~~So it is not an Achilles number.~~

	~~== References ==~~
	~~{{reflist}}~~


	~~{{Divisor classes}}~~
	~~{{Classes of natural numbers}}~~

	~~{{DEFAULTSORT:Achilles Number}}~~
	~~[[Category:Integer sequences]~~]

74.64.43.165 at 14:57, 7 January 2014

2014-01-07T14:57:11Z

← Older revision		Revision as of 16:57, 7 January 2014
Line 1:		Line 1:
	~~Greetings! I am Marvella and I really feel comfortable when people use~~ the ~~full title~~. ~~For years he~~'~~s been operating as a meter reader~~ and it'~~s some thing he truly appreciate~~. ~~What I adore performing is taking part in baseball~~ but ~~I haven~~'~~t produced~~ a ~~dime with it. South Dakota~~ is ~~where I~~'~~ve usually been living~~.<br><br>~~Also visit my webpage .~~.. [http://~~calvaryhill~~.net/xe/~~board_DsjO50~~/~~23648 calvaryhill~~.~~net~~]		An '''Achilles number''' is a number that is [[Powerful number\|powerful]] but not a [[perfect power]].<ref name=mw>{{MathWorld\|urlname=AchillesNumber\|title=Achilles Number}}
			</ref> A positive integer ''n'' is a powerful number if, for every [[prime factor]] ''p'' of ''n'', ''p''<sup>2</sup> is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as ''m<sup>k</sup>'', where ''m'' and ''k'' are positive integers greater than 1.

			Achilles numbers are named after [[Achilles]], a hero of the [[Trojan war]], who was also powerful but imperfect.

			==Sequence of Achilles numbers==
			A number ''n'' = ''p''<sub>1</sub><sup>''a''<sub>1</sub></sup>''p''<sub>2</sub><sup>''a''<sub>2</sub></sup>…''p''<sub>''k''</sub><sup>''a''<sub>''k''</sub></sup> is [[powerful number\|powerful]] if min(''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''k''</sub>) ≥ 2. If in addition gcd(''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''k''</sub>) = 1 the number is an Achilles number.

			The Achilles numbers up to 5000 are:
			:72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 {{OEIS\|id=A052486}}.

			The smallest pair of consecutive Achilles numbers is:<ref>Carlos Rivera, ''The Prime Puzzles and Problem Connection'', [http://www.primepuzzles.net/problems/prob_053.htm Problem 53]</ref>

			: 5425069447 = 7<sup>3</sup> × 41<sup>2</sup> × 97<sup>2</sup>
			: 5425069448 = 2<sup>3</sup> × 26041<sup>2</sup>

			==Examples==
			108 is a powerful number. Its [[prime factorization]] is 2<sup>2</sup> · 3<sup>3</sup>, and thus its prime factors are 2 and 3. Both 2<sup>2</sup> = 4 and 3<sup>2</sup> = 9 are divisors of 108. However, 108 cannot be represented as ''m<sup>k</sup>'', where ''m'' and ''k'' are positive integers greater than 1, so 108 is an Achilles number.

			Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 2<sup>2</sup> = 4 and 7<sup>2</sup> = 49 are divisors of it. Nonetheless, it is a perfect power:

			:<math>784=2^4 \cdot 7^2 = (2^2)^2 \cdot 7^2 = (2^2 \cdot 7)^2 = 28^2. \, </math>

			So it is not an Achilles number.

			== References ==
			{{reflist}}


			{{Divisor classes}}
			{{Classes of natural numbers}}

			{{DEFAULTSORT:Achilles Number}}
			[[Category:Integer sequences]]

en>Isheden: Improved explanation

2012-08-24T15:49:45Z

Improved explanation

New page

Greetings! I am Marvella and I really feel comfortable when people use the full title. For years he's been operating as a meter reader and it's some thing he truly appreciate. What I adore performing is taking part in baseball but I haven't produced a dime with it. South Dakota is where I've usually been living.<br><br>Also visit my webpage ... [http://calvaryhill.net/xe/board_DsjO50/23648 calvaryhill.net]

← Older revision		Revision as of 07:41, 11 December 2014
Line 1:		Line 1:
	~~The author is known by the name of Numbers Lint. One of the extremely very best issues in the globe for me is to do aerobics~~ and ~~now I'm attempting to earn cash with it~~. ~~Managing individuals is his profession. California is exactly where her house is but she needs to move simply because~~ of ~~her family~~.<br><br>~~Also visit~~ my ~~web page -~~ [http://~~torontocartridge~~.com/~~uncategorized/the-ideal-way-to-battle-a-yeast-infection~~/ ~~home std test~~]		Architectural, Building and Surveying Professionals Judson from Red Lake, enjoys skeet shooting, ganhando dinheiro na internet and sketching. Gains a lot of encouragement from life by touring spots like Sun Temple.<br><br>my blog :: [http://www.comoganhardinheiro101.com/inicio/ como ficar rico]