61 - Revision history

Vipul at 04:23, 16 January 2012

2012-01-16T04:23:51Z

← Older revision		Revision as of 04:23, 16 January 2012
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	The number 61 is a [[prime number]].		The number 61 is a [[prime number]].

			===Properties and families===

			{\| class="sortable" border="1"
			! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
			\|-
			\| [[satisfies property::prime number]] \|\| \|\| {{#lst:prime number\|list}} \|\| divide and check
			\|-
			\| [[satisfies property::regular prime]]\|\| \|\| {{#lst:regular prime\|list}} \|\|
			\|}

			==Waring representations==

			===Sums of squares===

			{{square sums facts to check against}}

			{\| class="sortable" border="1"
			! Item !! Value
			\|-
			\| unique (up to plus/minus and ordering) representation as sum of two squares \|\| <math>6^2 + 5^2</math>. Note that existence and uniqueness both follow from it being a prime that is 1 mod 4.<br>This also corresponds to the factorization <math>61 = (6 + 5i)(6 - 5i)</math> in the [[ring of Gaussian integers]] <math>\mathbb{Z}[i]</math>.
			\|-
			\| representations as sum of three squares (up to ordering and plus/minus equivalence) \|\| <math>6^2 + 5^2 + 0^2</math><br><math>6^2 + 4^2 + 3^2</math>
			\|}

	==Prime-generating polynomials==		==Prime-generating polynomials==

Vipul at 22:46, 15 January 2012

2012-01-15T22:46:43Z

← Older revision		Revision as of 22:46, 15 January 2012
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			{{particular natural number}}

	==Summary==		==Summary==

Vipul: /* Prime-generating polynomials */

2012-01-15T22:44:32Z

Prime-generating polynomials

← Older revision		Revision as of 22:44, 15 January 2012
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	\|-		\|-
	\| <math>n^2 - n + 41</math> \|\| 2 \|\| all numbers 1-40, because 41 is one of the [[lucky numbers of Euler]]. \|\| 5		\| <math>n^2 - n + 41</math> \|\| 2 \|\| all numbers 1-40, because 41 is one of the [[lucky numbers of Euler]]. \|\| 5
			\|-
			\| <math>2n^2 + 29</math> \|\| 2 \|\| all numbers 0-28 \|\| 4
	\|}		\|}

Vipul: Created page with "==Summary== ===Factorization=== The number 61 is a prime number. ==Prime-generating polynomials== Below are some polynomials that give prime numbers for small input va..."

2012-01-15T22:12:15Z

Created page with "==Summary== ===Factorization=== The number 61 is a prime number. ==Prime-generating polynomials== Below are some polynomials that give prime numbers for small input va..."

New page

==Summary==

===Factorization===

The number 61 is a [[prime number]].

==Prime-generating polynomials==

Below are some polynomials that give prime numbers for small input values, which give the value 61 for suitable input choice.

{| class="sortable" border="1"
! Polynomial !! Degree !! Some values for which it generates primes !! Input value <math>n</math> at which it generates 61
|-
| <math>n^2 - n + 41</math> || 2 || all numbers 1-40, because 41 is one of the [[lucky numbers of Euler]]. || 5
|}