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In [[number theory]], a '''Heegner number''' is a [[Square-free integer|square-free positive integer]] ''d'' such that the imaginary [[quadratic field]] '''Q'''({{sqrt|−''d''}}) has [[ideal class group|class number]] 1. Equivalently, its [[ring of integers]] has [[unique factorization]].<ref>{{cite book
  | last = Conway
  | first = John Horton
  | authorlink = John Horton Conway
  | coauthors = Guy, Richard K.
  | title = The Book of Numbers
  | publisher = Springer
  | year = 1996
  | page = 224
  | isbn = 0-387-97993-X }}
</ref>
 
The determination of such numbers is a special case of the [[class number problem]], and they underlie several striking results in number theory.
 
According to the [[Stark–Heegner theorem]] there are precisely nine Heegner numbers:
:{{num|1}}, {{num|2}}, {{num|3}}, {{num|7}}, {{num|11}}, {{num|19}}, {{num|43}}, {{num|67}}, {{num|163}}.
This result was conjectured by [[Carl Friedrich Gauss|Gauss]] and proven by [[Kurt Heegner]] in 1952.
 
==Euler's prime-generating polynomial==
Euler's [[Formula for primes#Prime formulas and polynomial functions|prime-generating polynomial]]
 
:<math>n^2 - n + 41, \, </math>,
 
which gives (distinct) primes for ''n''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;40, is related to the Heegner number 163&nbsp;=&nbsp;4&nbsp;·&nbsp;41&nbsp;−&nbsp;1.
 
Euler's formula, with <math>n</math> taking the values  1,... 40 is equivalent to
:<math>n^2 + n + 41, \, </math>
 
with <math>n</math> taking the values  0,... 39, and Rabinowitz<ref>Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math. (Cambridge) 1, 418–421, 1913.</ref> proved that
:<math>n^2 + n + p \, </math>
gives primes for <math>n=0,\dots,p-2</math> if and only if its discriminant <math>1-4p</math> equals minus a Heegner number.
 
(Note that <math>p-1</math> yields <math>p^2</math>, so <math>p-2</math> is maximal.)
1, 2, and 3 are not of the required form, so the Heegner numbers that work are <math>7, 11, 19, 43, 67, 163</math>, yielding prime generating functions of Euler's form for <math>2,3,5,11,17,41</math>; these latter numbers are called ''[[lucky numbers of Euler]]'' by [[François Le Lionnais|F. Le Lionnais]].<ref>Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983.</ref>
 
==Almost integers and Ramanujan's constant==
'''Ramanujan's constant''' is the [[transcendental number]]<ref>{{MathWorld|title=Transcendental Number|urlname=TranscendentalNumber}} gives <math>e^{\pi\sqrt{d}}, d \in Z^*</math>, based on
Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations." Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974.</ref>
<math>e^{\pi \sqrt{163}}</math>, which is an [[almost integer]], in that it is [[Mathematical coincidence#Containing pi or e and number 163|very close]] to an [[integer]]:
 
:<math>e^{\pi \sqrt{163}} = 262{,}537{,}412{,}640{,}768{,}743.99999999999925... </math><ref>[http://mathworld.wolfram.com/RamanujanConstant.html Ramanujan Constant – from Wolfram MathWorld<!-- Bot-generated title -->]</ref> <math>\approx 640320^3+744.</math>
 
This number was discovered in 1859 by the mathematician [[Charles Hermite]].<ref>{{cite book
  | last = Barrow
  | first = John D
  | title = The Constants of Nature
  | publisher = Jonathan Cape
  | year = 2002
  | location = London
  | isbn = 0-224-06135-6 }}
</ref>
In a 1975 [[April Fools' Day|April Fool]] article in ''[[Scientific American]]'' magazine,<ref>{{cite journal
  | last = Gardner
  | first = Martin
  | title = Mathematical Games
  | journal = Scientific American
  | volume = 232
  | issue = 4
  | page = 127
  | date = April 1975
  | publisher = Scientific American, Inc }}
</ref> "Mathematical Games" columnist [[Martin Gardner]] made the (hoax) claim that the number was in fact an integer, and that the Indian mathematical genius [[Srinivasa Ramanujan]] had predicted it—hence its name.
 
This coincidence is explained by [[complex multiplication]] and the [[q-expansion|''q''-expansion]] of the [[j-invariant]].
 
===Detail===
Briefly, <math>j((1+\sqrt{-d})/2)</math> is an integer for&nbsp;''d'' a Heegner number, and <math>e^{\pi \sqrt{d}} \approx -j((1+\sqrt{-d})/2) + 744</math> via the ''q''-expansion.
 
If <math>\tau</math> is a quadratic irrational, then the ''j''-invariant is an [[algebraic integer]] of degree <math>|\mbox{Cl}(\mathbf{Q}(\tau))|</math>, the [[Class number (number theory)|class number]] of <math>\mathbf{Q}(\tau)</math> and the minimal (monic integral) polynomial it satisfies is called the '''Hilbert class polynomial'''. Thus if the imaginary quadratic extension <math>\mathbf{Q}(\tau)</math> has class number 1 (so ''d'' is a Heegner number), the ''j''-invariant is an integer.
 
The [[Q-expansion|''q''-expansion]] of ''j'', with its [[Fourier series]] expansion written as a [[Laurent series]] in terms of <math>q=\exp(2 \pi i \tau)</math>, begins as:
 
:<math>j(q) = \frac{1}{q} + 744 + 196884 q + \cdots.</math><!-- don't put thousands separators under <math>! -->
 
The coefficients <math>c_n</math> asymptotically grow as <math>\ln(c_n) \sim 4\pi \sqrt{n} + O(\ln(n))</math>, and the low order coefficients grow more slowly than <math>200000^n</math>, so for <math>q \ll 1/200000</math>, ''j'' is very well approximated by its first two terms.  Setting <math>\tau = (1+\sqrt{-163})/2</math> yields <math>q=-\exp(-\pi \sqrt{163})</math> or equivalently, <math>\frac{1}{q}=-\exp(\pi \sqrt{163})</math>.  Now <math>j((1+\sqrt{-163})/2)=(-640320)^3</math>, so,
:<math>(-640320)^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right).</math><!-- don't put thousands separators under <math>! -->
Or,
:<math>e^{\pi \sqrt{163}}=640320^3+744+O\left(e^{-\pi \sqrt{163}}\right)</math>
where the linear term of the error is,
:<math>-196884/e^{\pi \sqrt{163}} \approx 196884/(640320^3+744)
\approx -0.00000000000075</math><!-- don't put thousands separators under <math>! -->
explaining why <math>e^{\pi \sqrt{163}}</math> is within approximately the above of being an integer.
 
==Other Heegner numbers==
For the four largest Heegner numbers, the approximations one obtains<ref>These can be checked by computing <math>\sqrt[3]{e^{\pi\sqrt{d}}-744}</math> on a calculator, and
<math>196884/e^{\pi\sqrt{d}}</math> for the linear term of the error.</ref> are as follows.
 
:<math>\begin{align}
e^{\pi \sqrt{19}}  &\approx 96^3+744-0.22\\
e^{\pi \sqrt{43}}  &\approx 960^3+744-0.00022\\
e^{\pi \sqrt{67}}  &\approx 5280^3+744-0.0000013\\
e^{\pi \sqrt{163}} &\approx 640320^3+744-0.00000000000075
\end{align}
</math><!-- don't put thousands separators under <math>! -->
 
Alternatively,<ref>http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#</ref>
:<math>\begin{align}
e^{\pi \sqrt{19}}  &\approx 12^3(3^2-1)^3+744-0.22\\
e^{\pi \sqrt{43}}  &\approx 12^3(9^2-1)^3+744-0.00022\\
e^{\pi \sqrt{67}}  &\approx 12^3(21^2-1)^3+744-0.0000013\\
e^{\pi \sqrt{163}} &\approx 12^3(231^2-1)^3+744-0.00000000000075
\end{align}
</math>
where the reason for the squares is due to certain [[Eisenstein series]].  For Heegner numbers <math>d < 19</math>, one does not obtain an almost integer; even <math>d = 19</math> is not noteworthy.<ref>The absolute deviation of a random real number (picked uniformly from [[unit interval|{{closed-closed|0,1|size=120%}}]], say) is a uniformly distributed variable on {{closed-closed|0, 0.5|size=120%}}, so it has [[absolute average deviation]] and [[median absolute deviation]] of 0.25, and a deviation of 0.22 is not exceptional.</ref> The integer ''j''-invariants are highly factorisable, which follows from the <math>12^3(n^2-1)^3=(2^2\cdot 3 \cdot (n-1) \cdot (n+1))^3</math> form, and factor as,
 
:<math>\begin{align}
j((1+\sqrt{-19})/2) &= 96^3 =(2^5 \cdot 3)^3\\
j((1+\sqrt{-43})/2) &= 960^3=(2^6 \cdot 3 \cdot 5)^3\\
j((1+\sqrt{-67})/2) & =5280^3=(2^5 \cdot 3 \cdot 5 \cdot 11)^3\\
j((1+\sqrt{-163})/2) &=640320^3=(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29)^3.
\end{align}
</math><!-- don't put thousands separators under <math>! -->
 
These [[transcendental numbers]], in addition to being closely approximated by integers, (which are simply [[algebraic numbers]] of degree&nbsp;1), can also be closely approximated by algebraic numbers of degree&nbsp;3,<ref>{{cite web|url=http://sites.google.com/site/tpiezas/001|title=Pi Formulas}}</ref>
 
:<math>\begin{align}
e^{\pi \sqrt{19}}  &\approx x^{24}-24; x^3-2x-2=0\\
e^{\pi \sqrt{43}}  &\approx x^{24}-24; x^3-2x^2-2=0\\
e^{\pi \sqrt{67}}  &\approx x^{24}-24; x^3-2x^2-2x-2=0\\
e^{\pi \sqrt{163}} &\approx x^{24}-24; x^3-6x^2+4x-2=0
\end{align}
</math>
 
The [[root of a function|roots]] of the cubics can be exactly given by quotients of the [[Dedekind eta function]] ''η''(''τ''), a modular function involving a 24th root, and which explains the 24 in the approximation.  In addition, they can also be closely approximated by algebraic numbers of degree 4,<ref>{{cite web|url=http://sites.google.com/site/tpiezas/ramanujan|title=Extending Ramanujan's Dedekind Eta Quotients}}</ref>
 
:<math>\begin{align}
e^{\pi \sqrt{19}}  &\approx 3^5 \left(3-\sqrt{2(-3+1\sqrt{3\cdot19})} \right)^{-2}-12.00006\dots\\
e^{\pi \sqrt{43}}  &\approx 3^5 \left(9-\sqrt{2(-39+7\sqrt{3\cdot43})} \right)^{-2}-12.000000061\dots\\
e^{\pi \sqrt{67}}  &\approx 3^5 \left(21-\sqrt{2(-219+31\sqrt{3\cdot67})} \right)^{-2}-12.00000000036\dots\\
e^{\pi \sqrt{163}}  &\approx 3^5 \left(231-\sqrt{2(-26679+2413\sqrt{3\cdot163})} \right)^{-2}-12.00000000000000021\dots
\end{align}
</math>
 
Note the reappearance of the integers <math>n = 3, 9, 21, 231</math> as well as the fact that,
 
:<math>\begin{align}
&2^6 \cdot 3(-3^2+3 \cdot 19 \cdot 1^2) = 96^2\\
&2^6 \cdot 3(-39^2+3 \cdot 43 \cdot 7^2) = 960^2\\
&2^6 \cdot 3(-219^2+3 \cdot 67 \cdot 31^2) = 5280^2\\
&2^6 \cdot 3(-26679^2+3 \cdot 163 \cdot 2413^2) = 640320^2
\end{align}
</math><!-- don't put thousands separators under <math>! -->
 
which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree&nbsp;6,
 
:<math>\begin{align}
e^{\pi \sqrt{19}}  &\approx (5x)^3-6.000010\dots\\
e^{\pi \sqrt{43}}  &\approx (5x)^3-6.000000010\dots\\
e^{\pi \sqrt{67}}  &\approx (5x)^3-6.000000000061\dots\\
e^{\pi \sqrt{163}} &\approx (5x)^3-6.000000000000000034\dots
\end{align}
</math>
 
where the ''x''s are given respectively by the appropriate root of the [[sextic equation]]s,
 
:<math>\begin{align}
&5x^6-96x^5-10x^3+1=0\\
&5x^6-960x^5-10x^3+1=0\\
&5x^6-5280x^5-10x^3+1=0\\
&5x^6-640320x^5-10x^3+1=0
\end{align}
</math><!-- don't put thousands separators under <math>! -->
 
with the j-invariants appearing again. These sextics are not only algebraic, they are also [[Solvable group|solvable]] in [[Nth root|radicals]] as they factor into two [[Cubic equation|cubics]] over the extension <math>\mathbb{Q}\sqrt{5}</math> (with the first factoring further into two [[Quadratic equation|quadratics]]). These algebraic approximations can be ''exactly'' expressed in terms of Dedekind eta quotients. As an example, let <math>\tau = (1+\sqrt{-163})/2</math>, then,
 
:<math>\begin{align}
e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/24} \eta(\tau)}{\eta(2\tau)} \right)^{24}-24.00000000000000105\dots\\
e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/12} \eta(\tau)}{\eta(3\tau)} \right)^{12}-12.00000000000000021\dots\\
e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/6} \eta(\tau)}{\eta(5\tau)} \right)^{6}-6.000000000000000034\dots
\end{align}
</math>
 
where the eta quotients are the algebraic numbers given above.
 
==Consecutive primes==
Given an odd prime&nbsp;''p'', if one computes <math>k^2 \pmod{p}</math> for <math>k=0,1,\dots,(p-1)/2</math> (this is sufficient because <math>(p-k)^2\equiv k^2 \pmod{p}</math>), one gets consecutive composites, followed by consecutive primes, if and only if ''p'' is a Heegner number.<ref>http://www.mathpages.com/home/kmath263.htm</ref>
 
For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.
 
==Notes and references==
<references />
 
==External links==
* {{MathWorld|title=Heegner Number|urlname=HeegnerNumber}}
* {{SloanesRef |sequencenumber=A003173|name=Heegner numbers: imaginary quadratic fields with unique factorization}}
* [http://www.ams.org/bull/1985-13-01/S0273-0979-1985-15352-2/S0273-0979-1985-15352-2.pdf Gauss' Class Number Problem for Imaginary Quadratic Fields, by Dorian Goldfeld]: Detailed history of problem.
* {{cite web|last=Clark|first=Alex|title=163 and Ramanujan Constant|url=http://www.numberphile.com/videos/163.html|work=Numberphile|publisher=[[Brady Haran]]}}
 
[[Category:Algebraic number theory]]

Revision as of 15:30, 21 December 2013

In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q(Template:Sqrt) has class number 1. Equivalently, its ring of integers has unique factorization.[1]

The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory.

According to the Stark–Heegner theorem there are precisely nine Heegner numbers:

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lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium, City Gate (Former KeyPoint) , a brand new upcoming Blended Growth that previously known as KeyPoint. This amazing improvement consist of 311 Residential and a complete of 188 commercial unit. An Iconic Masterpiece with three-storey of shopping and eating podium with the residential tower that rises above provides colour to the fast surrounding. A distinct icon that may rework the landscape of Seaside Rd.

lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium, City Gate (Former KeyPoint) , a brand new upcoming Blended Growth that previously known as KeyPoint. This amazing improvement consist of 311 Residential and a complete of 188 commercial unit. An Iconic Masterpiece with three-storey of shopping and eating podium with the residential tower that rises above provides colour to the fast surrounding. A distinct icon that may rework the landscape of Seaside Rd.

lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium, City Gate (Former KeyPoint) , a brand new upcoming Blended Growth that previously known as KeyPoint. This amazing improvement consist of 311 Residential and a complete of 188 commercial unit. An Iconic Masterpiece with three-storey of shopping and eating podium with the residential tower that rises above provides colour to the fast surrounding. A distinct icon that may rework the landscape of Seaside Rd.

lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium, City Gate (Former KeyPoint) , a brand new upcoming Blended Growth that previously known as KeyPoint. This amazing improvement consist of 311 Residential and a complete of 188 commercial unit. An Iconic Masterpiece with three-storey of shopping and eating podium with the residential tower that rises above provides colour to the fast surrounding. A distinct icon that may rework the landscape of Seaside Rd.

lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium, City Gate (Former KeyPoint) , a brand new upcoming Blended Growth that previously known as KeyPoint. This amazing improvement consist of 311 Residential and a complete of 188 commercial unit. An Iconic Masterpiece with three-storey of shopping and eating podium with the residential tower that rises above provides colour to the fast surrounding. A distinct icon that may rework the landscape of Seaside Rd.

lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium, City Gate (Former KeyPoint) , a brand new upcoming Blended Growth that previously known as KeyPoint. This amazing improvement consist of 311 Residential and a complete of 188 commercial unit. An Iconic Masterpiece with three-storey of shopping and eating podium with the residential tower that rises above provides colour to the fast surrounding. A distinct icon that may rework the landscape of Seaside Rd.

lengthy-term commitments to buy gathering companies. This seems to be the first The good news is, whereas the primary preview is private and usually reserved for administrators, business associates, and even shareholders of the developer, the first public preview is open to only about anyone with the means and want to purchase a unit. But you'd normally have to pre-register for it. Additionally, you will be registered for the VIP Preview should you so request. That sometimes entails submitting a kind listing your particulars & most well-liked units, together with your cheque. The agent will usually take care of a lot of the paperwork. Contact the agent to e mail you the necessary documents. Costs can be decrease than after the showflat is opened to stroll-ins. Upcoming co ndo In 2013 Octorber

Is also well connected to the remainder of Singapore by way of major expressways BKE and PIE. Including convenience even to the motorists. New launch FREEHOLD apartment Whitehaven residing by the west New launch rental Flora Ville New launch rental Lakefront living at Jurong Current property launch which are hidden gems Continuereading "The Crest Apartment by Wing Tai Preview Launch Soon" Current New Condo Launch Highlights Yahoo Singapore couldn't independently confirm the claims within the put up however has contacted SingTel for feedback. New launch rental 28 RC Suites New launch apartment metropolis fringe living by the Riverfront New launch condo Waterfront residing at Rivertrees Belgravia Villas Cluster House @ Ang Mo Kio Open for Sale 19September N ew Condo City Vista Tanah Merah Launch 2013 Full rental facilities

One other factor to think about when deciding between landed property and a Singapore new condos in singapore launches condominium is how shut you wish to be to different facilities like banks, hospitals, buying centers, public transportation, and so on. Landed houses as an entire are typically further away these constructions while condos are developed round these nodes for reasons of accessibility. If you happen to do not own a car, finding a condo would not be a bad concept. SGDeveloper.com showcase unique developer new properties launches in Singapore or abroad. Our workforce of skilled real property consultants from various companies will enable you in securing your new dream dwelling or in your property funding needs. Residents Save the Property 2012 Simply Launch Condominium New launch condominium combined growth by CDL

Lush Acres Govt Condominium is going to launch soon at Sengkang West Manner By Metropolis Developments Restricted. Lush Acres displays the broad expense of area, environmentally-inexperienced constructing and luxurious panorama of this low density growth with only 22% of website coverage. There have been college students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd really consisted of genuine patrons, or simply half-timers being paid to pretend how scorching the new challenge was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France on the INPI (National Institute of Intellectual Property) Coastal industrial spaceports

The Brownsville, Texas web site, therefore, gives an enticing combination of deep southern location, launch from the seaside, favorable regulatory environment, experienced workforce and ocean-delivery terminal. Clearly, SpaceX is exploring an alluring strategy of creating a sizeable personal business spaceport there. By Premium Developer, Wingtai Designed By Famend Architect, Toyo Ito (Vivo City, Belle Vue Residences, etc) Uncommon 256,000 sqft Condominium (Rectangular Land) Serene & Exclusive Atmosphere Quick Drive To Metropolis. Stroll To Redhill MRT Near Prestigious Schools Such As Crescent Girls' & Gan Eng Seng his house and simply past a stand of Oak and Ash, the hillside's natural form owned by Chesapeake. It also agreed to purchase industrial tools used to

Developer QingJian Realty Group Pte Ltd Address Woodlands Avenue 5 Area North Tenure ninety nine years Estimated Completion Web site Space 58,812 sq. metre Development Sort Executive Condominium Whole No. of Items Carpark Tons Amenities Unit Varieties That will soon change. The personal company Area Exploration Applied sciences ( SpaceX ) has been narrowing its search for a brand new location through which to construct a business launch facility for industrial clients. From its authentic listing, which included Virginia, Alaska and California, the corporate has not too long ago focused on Florida, Georgia, Puerto Rico and significantly Texas. Will SpaceX Launch from Texas? (Unique Video) District 20, Freehold Strata Cluster house District 10, Freehold house District 10, Freehold Condominium.

This result was conjectured by Gauss and proven by Kurt Heegner in 1952.

Euler's prime-generating polynomial

Euler's prime-generating polynomial

n2−n+41,,

which gives (distinct) primes for n = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1.

Euler's formula, with n taking the values 1,... 40 is equivalent to

n2+n+41,

with n taking the values 0,... 39, and Rabinowitz[2] proved that

n2+n+p

gives primes for n=0,…,p−2 if and only if its discriminant 1−4p equals minus a Heegner number.

(Note that p−1 yields p2, so p−2 is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7,11,19,43,67,163, yielding prime generating functions of Euler's form for 2,3,5,11,17,41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[3]

Almost integers and Ramanujan's constant

Ramanujan's constant is the transcendental number[4] eπ163, which is an almost integer, in that it is very close to an integer:

eπ163=262,537,412,640,768,743.99999999999925...[5] ≈6403203+744.

This number was discovered in 1859 by the mathematician Charles Hermite.[6] In a 1975 April Fool article in Scientific American magazine,[7] "Mathematical Games" columnist Martin Gardner made the (hoax) claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.

This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.

Detail

Briefly, j((1+−d)/2) is an integer for d a Heegner number, and eπd≈−j((1+−d)/2)+744 via the q-expansion.

If τ is a quadratic irrational, then the j-invariant is an algebraic integer of degree |Cl(Q(τ))|, the class number of Q(τ) and the minimal (monic integral) polynomial it satisfies is called the Hilbert class polynomial. Thus if the imaginary quadratic extension Q(τ) has class number 1 (so d is a Heegner number), the j-invariant is an integer.

The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of q=exp⁡(2πiτ), begins as:

j(q)=1q+744+196884q+⋯.

The coefficients cn asymptotically grow as ln⁡(cn)∼4πn+O(ln⁡(n)), and the low order coefficients grow more slowly than 200000n, so for q≪1/200000, j is very well approximated by its first two terms. Setting τ=(1+−163)/2 yields q=−exp⁡(−π163) or equivalently, 1q=−exp⁡(π163). Now j((1+−163)/2)=(−640320)3, so,

(−640320)3=−eπ163+744+O(e−π163).

Or,

eπ163=6403203+744+O(e−π163)

where the linear term of the error is,

−196884/eπ163≈196884/(6403203+744)≈−0.00000000000075

explaining why eπ163 is within approximately the above of being an integer.

Other Heegner numbers

For the four largest Heegner numbers, the approximations one obtains[8] are as follows.

eπ19≈963+744−0.22eπ43≈9603+744−0.00022eπ67≈52803+744−0.0000013eπ163≈6403203+744−0.00000000000075

Alternatively,[9]

eπ19≈123(32−1)3+744−0.22eπ43≈123(92−1)3+744−0.00022eπ67≈123(212−1)3+744−0.0000013eπ163≈123(2312−1)3+744−0.00000000000075

where the reason for the squares is due to certain Eisenstein series. For Heegner numbers d<19, one does not obtain an almost integer; even d=19 is not noteworthy.[10] The integer j-invariants are highly factorisable, which follows from the 123(n2−1)3=(22⋅3⋅(n−1)⋅(n+1))3 form, and factor as,

j((1+−19)/2)=963=(25⋅3)3j((1+−43)/2)=9603=(26⋅3⋅5)3j((1+−67)/2)=52803=(25⋅3⋅5⋅11)3j((1+−163)/2)=6403203=(26⋅3⋅5⋅23⋅29)3.

These transcendental numbers, in addition to being closely approximated by integers, (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,[11]

eπ19≈x24−24;x3−2x−2=0eπ43≈x24−24;x3−2x2−2=0eπ67≈x24−24;x3−2x2−2x−2=0eπ163≈x24−24;x3−6x2+4x−2=0

The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. In addition, they can also be closely approximated by algebraic numbers of degree 4,[12]

eπ19≈35(3−2(−3+13⋅19))−2−12.00006…eπ43≈35(9−2(−39+73⋅43))−2−12.000000061…eπ67≈35(21−2(−219+313⋅67))−2−12.00000000036…eπ163≈35(231−2(−26679+24133⋅163))−2−12.00000000000000021…

Note the reappearance of the integers n=3,9,21,231 as well as the fact that,

26⋅3(−32+3⋅19⋅12)=96226⋅3(−392+3⋅43⋅72)=960226⋅3(−2192+3⋅67⋅312)=5280226⋅3(−266792+3⋅163⋅24132)=6403202

which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree 6,

eπ19≈(5x)3−6.000010…eπ43≈(5x)3−6.000000010…eπ67≈(5x)3−6.000000000061…eπ163≈(5x)3−6.000000000000000034…

where the xs are given respectively by the appropriate root of the sextic equations,

5x6−96x5−10x3+1=05x6−960x5−10x3+1=05x6−5280x5−10x3+1=05x6−640320x5−10x3+1=0

with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension ℚ5 (with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let τ=(1+−163)/2, then,

eπ163=(eπi/24η(τ)η(2τ))24−24.00000000000000105…eπ163=(eπi/12η(τ)η(3τ))12−12.00000000000000021…eπ163=(eπi/6η(τ)η(5τ))6−6.000000000000000034…

where the eta quotients are the algebraic numbers given above.

Consecutive primes

Given an odd prime p, if one computes k2(modp) for k=0,1,…,(p−1)/2 (this is sufficient because (p−k)2≡k2(modp)), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.[13]

For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.

Notes and references

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math. (Cambridge) 1, 418–421, 1913.
  3. ↑ Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983.
  4. ↑

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  5. ↑ Ramanujan Constant – from Wolfram MathWorld
  6. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  8. ↑ These can be checked by computing eπd−7443 on a calculator, and 196884/eπd for the linear term of the error.
  9. ↑ http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#
  10. ↑ The absolute deviation of a random real number (picked uniformly from [[unit interval|Template:Closed-closed]], say) is a uniformly distributed variable on Template:Closed-closed, so it has absolute average deviation and median absolute deviation of 0.25, and a deviation of 0.22 is not exceptional.
  11. ↑ Template:Cite web
  12. ↑ Template:Cite web
  13. ↑ http://www.mathpages.com/home/kmath263.htm

External links



  • I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.



    Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
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  • Gauss' Class Number Problem for Imaginary Quadratic Fields, by Dorian Goldfeld: Detailed history of problem.
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