| Revision as of 04:24, 19 June 2015 edit137.82.118.155 (talk) review← Previous edit |
Revision as of 23:00, 27 July 2015 edit undoBG19bot (talk | contribs)1,005,055 editsm WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWBNext edit → |
| Line 1: |
Line 1: |
|
In number theory<!--mathematics-->, a '''Descartes number''' is an odd number which would have been an ], if one of its composite factors were prime. They are named after ] who observed that the number {{math| ''D'' {{=}} 3<sup>2</sup>⋅7<sup>2</sup>⋅11<sup>2</sup>⋅13<sup>2</sup>⋅22021 {{=}} (3⋅1001)<sup>2</sup>⋅(22⋅1001 − 1) {{=}} 198585576189 }} would be an ] if only {{math| 22021 }} were a ], since the ] for {{math| ''D'' }} would satisfy, if 22021 were prime, |
|
In number theory<!--mathematics-->, a '''Descartes number''' is an odd number which would have been an ], if one of its composite factors were prime. They are named after ] who observed that the number {{math| ''D'' {{=}} 3<sup>2</sup>⋅7<sup>2</sup>⋅11<sup>2</sup>⋅13<sup>2</sup>⋅22021 {{=}} (3⋅1001)<sup>2</sup>⋅(22⋅1001 − 1) {{=}} 198585576189 }} would be an ] if only {{math| 22021 }} were a ], since the ] for {{math| ''D'' }} would satisfy, if 22021 were prime, |
|
|
|
|
|
:<math>\begin{align} |
|
:<math>\begin{align} |
| Line 8: |
Line 8: |
|
\end{align}</math> |
|
\end{align}</math> |
|
|
|
|
|
where we turn a blind eye to the fact that {{math| 19<sup>2</sup>⋅61 {{=}} 22021 }} reveals that 22021 is ]! |
|
where we turn a blind eye to the fact that {{math| 19<sup>2</sup>⋅61 {{=}} 22021 }} reveals that 22021 is ]! |
|
|
|
|
|
A Descartes number is defined as an odd number {{math| ''n'' {{=}} ''m''⋅''p'' }} where {{math| ''m'' }} and {{math| ''p'' }} are ] and {{math| 2''n'' {{=}} σ(''m'')⋅(''p'' + 1) }}, whence {{math| ''p'' }} is taken as a 'spoof' prime. The example given is the only one currently known. |
|
A Descartes number is defined as an odd number {{math| ''n'' {{=}} ''m''⋅''p'' }} where {{math| ''m'' }} and {{math| ''p'' }} are ] and {{math| 2''n'' {{=}} σ(''m'')⋅(''p'' + 1) }}, whence {{math| ''p'' }} is taken as a 'spoof' prime. The example given is the only one currently known. |
|
|
|
|
|
If {{math| ''m'' }} is an odd ]<ref>Currently, the only known almost perfect numbers are the nonnegative powers of 2, whence the only known odd almost perfect number is {{math| 2<sup>0</sup> {{=}} 1. }}</ref>, that is, {{math| σ(''m'') {{=}} 2''m'' − 1 }} and {{math| 2''m'' − 1 }} is taken as a 'spoof' prime, then {{math| ''n'' {{=}} ''m''⋅(2''m'' − 1) }} is a Descartes number, since {{math| σ(''n'') {{=}} σ(''m''⋅(2''m'' − 1)) {{=}} σ(''m'')⋅2''m'' {{=}} (2''m'' − 1)⋅2''m'' {{=}} 2''n'' }}. If {{math| 2''m'' − 1 }} were prime, {{math| ''n'' }} would be an odd perfect number! |
|
If {{math| ''m'' }} is an odd ],<ref>Currently, the only known almost perfect numbers are the nonnegative powers of 2, whence the only known odd almost perfect number is {{math| 2<sup>0</sup> {{=}} 1. }}</ref> that is, {{math| σ(''m'') {{=}} 2''m'' − 1 }} and {{math| 2''m'' − 1 }} is taken as a 'spoof' prime, then {{math| ''n'' {{=}} ''m''⋅(2''m'' − 1) }} is a Descartes number, since {{math| σ(''n'') {{=}} σ(''m''⋅(2''m'' − 1)) {{=}} σ(''m'')⋅2''m'' {{=}} (2''m'' − 1)⋅2''m'' {{=}} 2''n'' }}. If {{math| 2''m'' − 1 }} were prime, {{math| ''n'' }} would be an odd perfect number! |
|
|
|
|
|
==Notes== |
|
==Notes== |
| Line 20: |
Line 20: |
|
* {{cite book | last1=Banks | first1=William D. | last2=Güloğlu | first2=Ahmet M. | last3=Nevans | first3=C. Wesley | last4=Saidak | first4=Filip | chapter=Descartes numbers | pages=167–173 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11004 }} |
|
* {{cite book | last1=Banks | first1=William D. | last2=Güloğlu | first2=Ahmet M. | last3=Nevans | first3=C. Wesley | last4=Saidak | first4=Filip | chapter=Descartes numbers | pages=167–173 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11004 }} |
|
* {{cite book | last1=Klee | first1=Victor | author-link1=Victor Klee | last2=Wagon | first2=Stan | author-link2=Stan Wagon | title=Old and new unsolved problems in plane geometry and number theory | series=The Dolciani Mathematical Expositions | volume=11 | location=Washington, DC | publisher=] | year=1991 | isbn=0-88385-315-9 | zbl=0784.51002 }} |
|
* {{cite book | last1=Klee | first1=Victor | author-link1=Victor Klee | last2=Wagon | first2=Stan | author-link2=Stan Wagon | title=Old and new unsolved problems in plane geometry and number theory | series=The Dolciani Mathematical Expositions | volume=11 | location=Washington, DC | publisher=] | year=1991 | isbn=0-88385-315-9 | zbl=0784.51002 }} |
|
|
|
|
⚫ |
{{Classes of natural numbers}} |
|
|
|
|
|
] |
|
] |
|
] |
|
] |
|
|
|
|
|
|
|
| ⚫ |
{{Classes of natural numbers}} |
|
|
{{numtheory-stub}} |
|
{{numtheory-stub}} |
where we turn a blind eye to the fact that 19⋅61 = 22021 reveals that 22021 is composite!
