Lucky number of Euler - Revision history

Vipul: /* Occurrence */

2012-01-15T22:26:55Z

Occurrence

← Older revision		Revision as of 22:26, 15 January 2012
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	There are only six lucky numbers of Euler:		There are only six lucky numbers of Euler:

	<section begin="list"/>[[2]], [[3]], [[5]], [[11]], [[17]], [[41]] [[Oeis:A014556\|View list of OEIS]]<section end="list"/>		<section begin="list"/>[[2]], [[3]], [[5]], [[11]], [[17]], [[41]] [[Oeis:A014556\|View list on OEIS]]<section end="list"/>

	==Polynomial values==		==Polynomial values==

Vipul: /* Polynomial values */

2012-01-15T22:26:33Z

Polynomial values

← Older revision		Revision as of 22:26, 15 January 2012
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	The table should eventually go up to <math>n = 40</math>. It is not yet completed.		The table should eventually go up to <math>n = 40</math>. It is not yet completed.

			Note that among the primes till 100, the only primes that are ''not'' covered in this table at least once are [[79]] and [[89]].

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul at 22:24, 15 January 2012

2012-01-15T22:24:18Z

← Older revision		Revision as of 22:24, 15 January 2012
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	<section begin="list"/>[[2]], [[3]], [[5]], [[11]], [[17]], [[41]] [[Oeis:A014556\|View list of OEIS]]<section end="list"/>		<section begin="list"/>[[2]], [[3]], [[5]], [[11]], [[17]], [[41]] [[Oeis:A014556\|View list of OEIS]]<section end="list"/>

			==Polynomial values==

			The table should eventually go up to <math>n = 40</math>. It is not yet completed.

			{\| class="sortable" border="1"
			! <math>n</math> !! <math>n^2 - n</math> !! Polynomial value <math>n^2 - n + 2</math> !! Polynomial value <math>n^2 - n + 3</math> !! Polynomial value <math>n^2 - n + 5</math> !! Polynomial value <math>n^2 - n + 11</math> !! Polynomial value <math>n^2 - n + 17</math> !! Polynomial value <math>n^2 - n + 41</math>
			\|-
			\| 1 \|\| 0 \|\| [[2]] \|\| [[3]] \|\| [[5]] \|\| [[11]] \|\| [[17]] \|\| [[41]]
			\|-
			\| 2 \|\| 2 \|\| N/A \|\| [[5]] \|\| [[7]] \|\| [[13]] \|\| [[19]] \|\| [[43]]
			\|-
			\| 3 \|\| 6 \|\| N/A \|\| N/A \|\| [[13]] \|\| [[19]] \|\| [[23]] \|\| [[47]]
			\|-
			\| 4 \|\| 12 \|\| N/A \|\| N/A \|\| [[17]] \|\| [[23]] \|\| [[29]] \|\| [[53]]
			\|-
			\| 5 \|\| 20 \|\| N/A \|\| N/A \|\| N/A \|\| [[31]] \|\| [[37]] \|\| [[61]]
			\|-
			\| 6 \|\| 30 \|\| N/A \|\| N/A \|\| N/A \|\| [[41]] \|\| [[47]] \|\| [[71]]
			\|-
			\| 7 \|\| 42 \|\| N/A \|\| N/A \|\| N/A \|\| [[53]] \|\| [[59]] \|\| [[83]]
			\|-
			\| 8 \|\| 56 \|\| N/A \|\| N/A \|\| N/A \|\| [[67]] \|\| [[73]] \|\| [[97]]
			\|-
			\| 9 \|\| 72 \|\| N/A \|\| N/A \|\| N/A \|\| [[83]] \|\| [[89]] \|\| [[113]]
			\|-
			\| 10 \|\| 90 \|\| N/A \|\| N/A \|\| N/A \|\| [[101]] \|\| [[107]] \|\| [[131]]
			\|-
			\| 11 \|\| 110 \|\| N/A \|\| N/A \|\| N/A \|\| N/A \|\| [[127]] \|\| [[151]]
			\|-
			\| 12 \|\| 132 \|\| N/A \|\| N/A \|\| N/A \|\| N/A \|\| [[149]] \|\| [[173]]
			\|}

Vipul: /* Definition */

2012-01-15T21:41:24Z

Definition

← Older revision		Revision as of 21:41, 15 January 2012
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	takes [[prime number]] values for <math>n \in \{ 1,2,\dots,p-1 \}</math>.		takes [[prime number]] values for <math>n \in \{ 1,2,\dots,p-1 \}</math>.

	This condition is equivalent to the condition that the ring of integers in <math>\mathbb{Q}(\sqrt{-1 - 4p})</math> be a unique factorization domain, or equivalently, the class number of the field be equal to one.		This condition is equivalent to the condition that the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> be a unique factorization domain, or equivalently, the class number of the field be equal to one.

	==Occurrence==		==Occurrence==

Vipul: Created page with "==Definition== A '''lucky number of Euler''' is a prime number $p$ such that the polynomial: $n^2 - n + p$ takes prime number values for
2012-01-15T21:41:13Z

Created page with "==Definition== A '''lucky number of Euler''' is a prime number <math>p</math> such that the polynomial: <math>n^2 - n + p</math> takes prime number values for <math..."

New page
==Definition==

A '''lucky number of Euler''' is a [[prime number]] <math>p</math> such that the polynomial:

<math>n^2 - n + p</math>

takes [[prime number]] values for <math>n \in \{ 1,2,\dots,p-1 \}</math>.

This condition is equivalent to the condition that the ring of integers in <math>\mathbb{Q}(\sqrt{-1 - 4p})</math> be a unique factorization domain, or equivalently, the class number of the field be equal to one.

==Occurrence==

There are only six lucky numbers of Euler:

<section begin="list"/>[[2]], [[3]], [[5]], [[11]], [[17]], [[41]] [[Oeis:A014556|View list of OEIS]]<section end="list"/>