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<div>In [[mathematics]], in the field of [[number theory]], the '''Ramanujan–Nagell equation''' is a particular [[exponential Diophantine equation]].<br />
<br />
==Equation and solution==<br />
The equation is<br />
:<math>2^n-7=x^2 \,</math><br />
and solutions in natural numbers ''n'' and ''x'' exist just when ''n'' = 3, 4, 5, 7 and 15.<br />
<br />
This was conjectured in 1913 by Indian mathematician [[Srinivasa Ramanujan]], proposed independently in 1943 by the Norwegian mathematician [[Wilhelm Ljunggren]], and [[mathematical proof|proved]] in 1948 by the Norwegian mathematician [[Trygve Nagell]]. The values on ''n'' correspond to the values of ''x'' as:-<br />
<br />
:''x'' = 1, 3, 5, 11 and 181.<ref>Saradha & Srinivasan (2008) p.208</ref><br />
<br />
== Triangular Mersenne numbers ==<br />
The problem of finding all numbers of the form 2<sup>''b''</sup>&nbsp;&minus;&nbsp;1 ([[Mersenne number]]s) which are [[triangular number|triangular]] is equivalent:<br />
<br />
:<math>2^b-1 = \frac{y(y+1)}{2}</math><br />
:<math>\Leftrightarrow 8(2^b-1) = 4y(y+1)</math><br />
:<math>\Leftrightarrow 2^{b+3}-8 = 4y^2+4y</math><br />
:<math>\Leftrightarrow 2^{b+3}-7 = 4y^2+4y+1</math><br />
:<math>\Leftrightarrow 2^{b+3}-7 = (2y+1)^2</math><br />
<br />
The values of ''b'' are just those of ''n''&nbsp;&minus;&nbsp;3, and the corresponding triangular Mersenne numbers (also known as '''Ramanujan–Nagell numbers''') are:<br />
<br />
:<math>\frac{y(y+1)}{2} = \frac{(x-1)(x+1)}{8}</math><br />
<br />
for ''x'' = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more {{OEIS|id=A076046}}.<br />
<br />
==Equations of Ramanujan–Nagell type==<br />
An equation of the form<br />
<br />
:<math> x^2 + D = A B^n </math><br />
<br />
for fixed ''D'', ''A'' , ''B'' and variable ''x'', ''n'' is said to be of ''Ramanujan–Nagell type''. A result of [[Carl Ludwig Siegel|Siegel]] implies that the number of solutions in each case is finite.<ref>Saradha & Srinivasan (2008) p.207</ref> The equation with ''A''=1, ''B''=2 has at most two solutions except in the case ''D''=7 already mentioned. There are infinitely many values of ''D'' for which there are two solutions, including <math>D = 2^m - 1</math>.<ref>Saradha & Srinivasan (2008) p.208</ref><br />
<br />
==Equations of Lebesgue–Nagell type==<br />
An equation of the form<br />
<br />
:<math> x^2 + D = A y^n </math><br />
<br />
for fixed ''D'', ''A'' and variable ''x'', ''y'', ''n'' is said to be of ''Lebesgue–Nagell type''. Results of Shorey and [[Robert Tijdeman|Tijdeman]] imply that the number of solutions in each case is finite.<ref>Saradha & Srinivasan (2008) p.211</ref><br />
<br />
==See also==<br />
* [[Scientific equations named after people]]<br />
<br />
==References==<br />
{{reflist}}<br />
* {{cite journal | author=S. Ramanujan | authorlink=Srinivasa Ramanujan | title=Question 464 | journal=J. Indian Math. Soc. | volume=5 | year=1913 | pages=130 }}<br />
* {{cite journal | author=W. Ljunggren | authorlink=WIlhelm Ljunggren | title=Oppgave nr 2 | journal=Norsk Mat. Tidsskr. | volume=25 | year=1943 | pages=29 }}<br />
* {{cite journal | author=T. Nagell | title=Løsning till oppgave nr 2 | journal=Norsk Mat. Tidsskr. | volume=30 | year=1948 | pages=62–64 }}<br />
* {{cite journal | author=T. Nagell | title=The Diophantine equation ''x''<sup>2</sup>+7=2<sup>''n''</sup> | journal=Ark. Mat. | volume=30 | year=1961 | pages=185–187 | doi=10.1007/BF02592006 }}<br />
* {{cite book | last1=Shorey | first1=T.N. | last2=Tijdeman | first2=R. | author2-link=Robert Tijdeman | title=Exponential Diophantine equations | series=Cambridge Tracts in Mathematics | volume=87 | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-26826-5 | zbl=0606.10011 | pages=137–138 }}<br />
* {{cite book | editor-first=N. | editor-last=Saradha | title=Diophantine Equations | publisher=Narosa | year=2008 | isbn=978-81-7319-898-4 | first1=N. | last1=Saradha | first2=Anitha | last2=Srinivasan | chapter=Generalized Lebesgue–Ramanujan–Nagell equations | pages=207–223 }}<br />
<br />
==External links==<br />
* {{cite web | url=http://mathworld.wolfram.com/RamanujansSquareEquation.html | publisher=Wolfram MathWorld | title=Values of X corresponding to N in the Ramanujan–Nagell Equation | accessdate=2012-05-08 }}<br />
* [http://mathforum.org/kb/message.jspa?messageID=419063&tstart=0 Can ''N''<sup>2</sup> + ''N'' + 2 Be A Power Of&nbsp;2?], Math Forum discussion<br />
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{{DEFAULTSORT:Ramanujan-Nagell equation}}<br />
[[Category:Diophantine equations]]<br />
[[Category:Srinivasa Ramanujan]]</div>92.40.78.0