Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
No edit summary
No edit summary
 
(313 intermediate revisions by more than 100 users not shown)
Line 1: Line 1:
In [[mathematics]], a '''congruent number''' is a positive [[integer]] that is the area of a [[right triangle]] with three [[rational number]] sides.<ref>{{MathWorld |urlname=CongruentNumber |title=Congruent Number}}</ref>  A more general definition includes all positive rational numbers with this property.<ref name=Koblitz>{{cite book |first = Neal |last = Koblitz |authorlink = Neal Koblitz |title = Introduction to Elliptic Curves and Modular Forms |isbn = 0-387-97966-2 |publisher = [[Springer-Verlag]] |year = 1993| page = 3 |location = New York}}</ref>
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


The sequence of integer congruent numbers starts with
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
: 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, … {{OEIS|A003273}}
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


For example, 5 is a congruent number because it is the area of a 20/3, 3/2, 41/6 triangle. Similarly, 6 is a congruent number because it is the area of a 3,4,5 triangle. 3 is not a congruent number.
Registered users will be able to choose between the following three rendering modes:


If ''q'' is a congruent number then ''s''<sup>2</sup>''q'' is also a congruent number for any rational number ''s'' (just by multiplying each side of the triangle by ''s'').  This leads to the observation that whether a nonzero rational number ''q'' is a congruent number depends only on its residue in the [[group (mathematics)|group]]
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math>\mathbb{Q}^{*}/\mathbb{Q}^{*2}</math>.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Every residue class in this group contains exactly one [[square free]] integer, and it is common, therefore, only to consider square free positive integers, when speaking about congruent numbers.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Congruent number problem==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
The question of determining whether a given rational number is a congruent number is called the '''congruent number problem'''. This problem has not (as of 2012) been brought to a successful resolution. [[Tunnell's theorem]] provides an easily testable criterion for determining whether a number is congruent; but his result relies on the [[Birch and Swinnerton-Dyer conjecture]], which is still unproven.


'''Fermat's right triangle theorem''', named after [[Pierre de Fermat]], states that no [[square number]] can be a congruent number.
==Demos==


==Relation to elliptic curves==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
The question of whether a given number is congruent turns out to be equivalent to the condition that a certain [[elliptic curve]] has positive [[rank of an abelian group|rank]].<ref name=Koblitz />  An alternative approach to the idea is presented below (as can essentially also be found in the introduction to Tunnell's paper).


Suppose ''a'',''b'',''c'' are numbers (not necessarily positive or rational) which satisfy the following two equations:


:<math>
* accessibility:
\begin{matrix}
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
a^2 + b^2 &=& c^2\\
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
\frac{ab}{2} &=& n.
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
\end{matrix}
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
</math>
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Then set ''x'' = ''n''(''a''+''c'')/''b'' and
==Test pages ==
''y'' = 2''n''<sup>2</sup>(''a''+''c'')/''b''<sup>2</sup>.
A calculation shows
:<math>
y^2 = x^3 -n^2x
\,\!
</math>
and ''y'' is not 0 (if ''y'' = 0 then ''a'' = -''c'', so ''b'' = 0, but (1/2)''ab'' = ''n'' is nonzero, a contradiction).


Conversely, if ''x'' and ''y'' are numbers which satisfy the above equation and ''y'' is not 0, set
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
''a'' = (''x''<sup>2</sup> - ''n''<sup>2</sup>)/''y'',
*[[Displaystyle]]
''b'' = 2''nx''/''y'', and ''c'' = (''x''<sup>2</sup> + ''n''<sup>2</sup>)/''y'' .  A calculation shows these three numbers
*[[MathAxisAlignment]]
satisfy the two equations for ''a'', ''b'', and ''c'' above.
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


These two correspondences between (''a'',''b'',''c'') and (''x'',''y'') are inverses of each other, so
*[[Inputtypes|Inputtypes (private Wikis only)]]
we have a one-to-one correspondence between any solution of the two equations in
*[[Url2Image|Url2Image (private Wikis only)]]
''a'', ''b'', and ''c'' and any solution of the equation in ''x'' and ''y'' with ''y'' nonzero.  In particular,
==Bug reporting==
from the formulas in the two correspondences, for rational ''n'' we see that ''a'', ''b'', and ''c'' are
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
rational if and only if the corresponding ''x'' and ''y'' are rational, and vice versa.
(We also have that ''a'', ''b'', and ''c'' are all positive if and only if ''x'' and ''y'' are all positive;
notice from the equation ''y''<sup>2</sup> = ''x''<sup>3</sup> - ''xn''<sup>2</sup> = ''x''(''x''<sup>2</sup> - ''n''<sup>2</sup>)
that if ''x'' and ''y'' are positive then ''x''<sup>2</sup> - ''n''<sup>2</sup> must be positive, so the formula for
''a'' above is positive.)
 
Thus a positive rational number ''n'' is congruent if and only if the equation
''y''<sup>2</sup> = ''x''<sup>3</sup> - ''n''<sup>2</sup>''x'' has a [[rational point]] with ''y'' not equal to 0.
It can be shown (as a nice application of [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]] on primes in arithmetic progression)
that the only torsion points on this elliptic curve are those with ''y'' equal to 0, hence the
existence of a rational point with ''y'' nonzero is equivalent to saying the elliptic curve has positive rank.
 
==Current progress==
Much work has been done classifying congruent numbers.
 
For example, it is known<ref>{{cite journal |author=[[Paul Monsky]] |title=Mock Heegner Points and Congruent Numbers |journal=Mathematische Zeitschrift |volume=204 |issue=1 |year=1990 |pages=45–67 |doi=10.1007/BF02570859}}</ref> that if ''p'' is a prime number then
*if ''p'' ≡ 3 ([[modular arithmetic|mod]] 8), then ''p'' is not a congruent number, but 2''p'' is a congruent number.
*if ''p'' ≡ 5 (mod 8), then ''p'' is a congruent number.
*if ''p'' ≡ 7 (mod 8), then ''p'' and 2''p'' are congruent numbers.
 
==References==
{{reflist}}
*A short discussion of the current state of the problem with many references can be found in [[Alice Silverberg]]'s [http://www.math.uci.edu/~asilverb/bibliography/pcmibook.ps Open Questions in Arithmetic Algebraic Geometry] (Postscript).
*Many references are given in {{cite book |author=[[Richard Guy]] |title=Unsolved Problems in Number Theory |isbn=0-387-20860-7}}
*For a history of the problem, see {{cite book |author=[[Leonard Eugene Dickson]] |title=History of the Theory of Numbers |volume=Volume II |isbn=0-8218-1935-6 |chapter=Chapter XVI}}
*{{cite journal |author=[[Ronald Alter]] |title=The Congruent Number Problem |journal=American Mathematical Monthly |volume=87 |issue=1 |year=1980 |pages=43–45 |doi=10.2307/2320381 |publisher=Mathematical Association of America |jstor=2320381}}
*{{cite journal |doi=10.1007/BF02837344 |author=V. Chandrasekar |title=The Congruent Number Problem |journal=Resonance |volume=3 |issue=8 |year=1998 |pages=33–45 |url=http://www.ias.ac.in/resonance/Aug1998/pdf/Aug1998p33-45.pdf}}
* {{cite journal
  | author = Tunnell, Jerrold B.
  | title = A classical Diophantine problem and modular forms of weight 3/2
  | journal = [[Inventiones Mathematicae]]
  | volume = 72
  | issue = 2
  | pages = 323–334
  | year = 1983
  | doi = 10.1007/BF01389327}}
* [http://www.aimath.org/news/congruentnumbers/ A Trillion Triangles] - mathematicians have resolved the first one trillion cases (conditional on the [[Birch and Swinnerton-Dyer conjecture]]).
 
[[Category:Arithmetic problems of plane geometry]]
[[Category:Unsolved problems in mathematics]]
[[Category:Number theory]]
[[Category:Elliptic curves]]
[[Category:Triangle geometry]]
 
[[he:מספר קונגרואנטי]]
[[ja:合同数]]
[[pt:Números congruentes]]
[[ru:Конгруэнтное число]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=329754"