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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
| | Eusebio Stanfill is what's blogged on my birth records although it is n't the name on particular birth certificate. Vermont can be where my home was. Software making has been my 24-hour period job for a despite. To farrenheit is the only hobby my wife doesn't agree to. You can believe my website here: http://circuspartypanama.com<br><br>Stop by my webpage: [http://circuspartypanama.com clash of clans hack tool] |
| | last = Conway
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| | first = John Horton
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| | authorlink = John Horton Conway
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| | coauthors = Guy, Richard K.
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| | title = The Book of Numbers
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| | publisher = Springer
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| | year = 1996
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| | page = 224
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| | isbn = 0-387-97993-X }}
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| </ref>
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| The determination of such numbers is a special case of the [[class number problem]], and they underlie several striking results in number theory.
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| According to the [[Stark–Heegner theorem]] there are precisely nine Heegner numbers:
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| :{{num|1}}, {{num|2}}, {{num|3}}, {{num|7}}, {{num|11}}, {{num|19}}, {{num|43}}, {{num|67}}, {{num|163}}.
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| This result was conjectured by [[Carl Friedrich Gauss|Gauss]] and proven by [[Kurt Heegner]] in 1952.
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| ==Euler's prime-generating polynomial==
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| Euler's [[Formula for primes#Prime formulas and polynomial functions|prime-generating polynomial]]
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| :<math>n^2 - n + 41, \, </math>,
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| which gives (distinct) primes for ''n'' = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1.
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| Euler's formula, with <math>n</math> taking the values 1,... 40 is equivalent to
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| :<math>n^2 + n + 41, \, </math>
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| 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
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| :<math>n^2 + n + p \, </math>
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| 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.
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| (Note that <math>p-1</math> yields <math>p^2</math>, so <math>p-2</math> is maximal.)
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| 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>
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| ==Almost integers and Ramanujan's constant==
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| '''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
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| 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>
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| <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]]:
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| :<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>
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| This number was discovered in 1859 by the mathematician [[Charles Hermite]].<ref>{{cite book
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| | last = Barrow
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| | first = John D
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| | title = The Constants of Nature
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| | publisher = Jonathan Cape
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| | year = 2002
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| | location = London
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| | isbn = 0-224-06135-6 }}
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| </ref>
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| In a 1975 [[April Fools' Day|April Fool]] article in ''[[Scientific American]]'' magazine,<ref>{{cite journal
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| | last = Gardner
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| | first = Martin
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| | title = Mathematical Games
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| | journal = Scientific American
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| | volume = 232
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| | issue = 4
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| | page = 127
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| | date = April 1975
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| | publisher = Scientific American, Inc }}
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| </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.
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| This coincidence is explained by [[complex multiplication]] and the [[q-expansion|''q''-expansion]] of the [[j-invariant]].
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| ===Detail===
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| Briefly, <math>j((1+\sqrt{-d})/2)</math> is an integer for ''d'' a Heegner number, and <math>e^{\pi \sqrt{d}} \approx -j((1+\sqrt{-d})/2) + 744</math> via the ''q''-expansion.
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| 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.
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| 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:
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| :<math>j(q) = \frac{1}{q} + 744 + 196884 q + \cdots.</math><!-- don't put thousands separators under <math>! -->
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| 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,
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| :<math>(-640320)^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right).</math><!-- don't put thousands separators under <math>! -->
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| Or,
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| :<math>e^{\pi \sqrt{163}}=640320^3+744+O\left(e^{-\pi \sqrt{163}}\right)</math>
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| where the linear term of the error is,
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| :<math>-196884/e^{\pi \sqrt{163}} \approx 196884/(640320^3+744)
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| \approx -0.00000000000075</math><!-- don't put thousands separators under <math>! -->
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| explaining why <math>e^{\pi \sqrt{163}}</math> is within approximately the above of being an integer.
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| ==Other Heegner numbers==
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| 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
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| <math>196884/e^{\pi\sqrt{d}}</math> for the linear term of the error.</ref> are as follows.
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| :<math>\begin{align}
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| e^{\pi \sqrt{19}} &\approx 96^3+744-0.22\\
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| e^{\pi \sqrt{43}} &\approx 960^3+744-0.00022\\
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| e^{\pi \sqrt{67}} &\approx 5280^3+744-0.0000013\\
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| e^{\pi \sqrt{163}} &\approx 640320^3+744-0.00000000000075
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| \end{align}
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| </math><!-- don't put thousands separators under <math>! -->
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| Alternatively,<ref>http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#</ref>
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| :<math>\begin{align}
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| e^{\pi \sqrt{19}} &\approx 12^3(3^2-1)^3+744-0.22\\
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| e^{\pi \sqrt{43}} &\approx 12^3(9^2-1)^3+744-0.00022\\
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| e^{\pi \sqrt{67}} &\approx 12^3(21^2-1)^3+744-0.0000013\\
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| e^{\pi \sqrt{163}} &\approx 12^3(231^2-1)^3+744-0.00000000000075
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| \end{align}
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| </math>
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| 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,
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| :<math>\begin{align}
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| j((1+\sqrt{-19})/2) &= 96^3 =(2^5 \cdot 3)^3\\
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| j((1+\sqrt{-43})/2) &= 960^3=(2^6 \cdot 3 \cdot 5)^3\\
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| j((1+\sqrt{-67})/2) & =5280^3=(2^5 \cdot 3 \cdot 5 \cdot 11)^3\\
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| j((1+\sqrt{-163})/2) &=640320^3=(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29)^3.
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| \end{align}
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| </math><!-- don't put thousands separators under <math>! -->
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| 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,<ref>{{cite web|url=http://sites.google.com/site/tpiezas/001|title=Pi Formulas}}</ref>
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| :<math>\begin{align} | |
| e^{\pi \sqrt{19}} &\approx x^{24}-24; x^3-2x-2=0\\
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| e^{\pi \sqrt{43}} &\approx x^{24}-24; x^3-2x^2-2=0\\
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| e^{\pi \sqrt{67}} &\approx x^{24}-24; x^3-2x^2-2x-2=0\\
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| e^{\pi \sqrt{163}} &\approx x^{24}-24; x^3-6x^2+4x-2=0
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| \end{align}
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| </math>
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| 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>
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| :<math>\begin{align}
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| e^{\pi \sqrt{19}} &\approx 3^5 \left(3-\sqrt{2(-3+1\sqrt{3\cdot19})} \right)^{-2}-12.00006\dots\\
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| e^{\pi \sqrt{43}} &\approx 3^5 \left(9-\sqrt{2(-39+7\sqrt{3\cdot43})} \right)^{-2}-12.000000061\dots\\
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| e^{\pi \sqrt{67}} &\approx 3^5 \left(21-\sqrt{2(-219+31\sqrt{3\cdot67})} \right)^{-2}-12.00000000036\dots\\
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| e^{\pi \sqrt{163}} &\approx 3^5 \left(231-\sqrt{2(-26679+2413\sqrt{3\cdot163})} \right)^{-2}-12.00000000000000021\dots
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| \end{align}
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| </math>
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| Note the reappearance of the integers <math>n = 3, 9, 21, 231</math> as well as the fact that,
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| :<math>\begin{align}
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| &2^6 \cdot 3(-3^2+3 \cdot 19 \cdot 1^2) = 96^2\\
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| &2^6 \cdot 3(-39^2+3 \cdot 43 \cdot 7^2) = 960^2\\
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| &2^6 \cdot 3(-219^2+3 \cdot 67 \cdot 31^2) = 5280^2\\
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| &2^6 \cdot 3(-26679^2+3 \cdot 163 \cdot 2413^2) = 640320^2
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| \end{align}
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| </math><!-- don't put thousands separators under <math>! --> | |
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| which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree 6,
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| :<math>\begin{align}
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| e^{\pi \sqrt{19}} &\approx (5x)^3-6.000010\dots\\
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| e^{\pi \sqrt{43}} &\approx (5x)^3-6.000000010\dots\\
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| e^{\pi \sqrt{67}} &\approx (5x)^3-6.000000000061\dots\\
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| e^{\pi \sqrt{163}} &\approx (5x)^3-6.000000000000000034\dots
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| \end{align}
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| </math>
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| where the ''x''s are given respectively by the appropriate root of the [[sextic equation]]s,
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| :<math>\begin{align} | |
| &5x^6-96x^5-10x^3+1=0\\
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| &5x^6-960x^5-10x^3+1=0\\
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| &5x^6-5280x^5-10x^3+1=0\\
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| &5x^6-640320x^5-10x^3+1=0
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| \end{align}
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| </math><!-- don't put thousands separators under <math>! -->
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| 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,
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| :<math>\begin{align}
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| e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/24} \eta(\tau)}{\eta(2\tau)} \right)^{24}-24.00000000000000105\dots\\
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| e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/12} \eta(\tau)}{\eta(3\tau)} \right)^{12}-12.00000000000000021\dots\\
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| e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/6} \eta(\tau)}{\eta(5\tau)} \right)^{6}-6.000000000000000034\dots
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| \end{align}
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| </math>
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| where the eta quotients are the algebraic numbers given above.
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| ==Consecutive primes==
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| Given an odd prime ''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>
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| For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.
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| ==Notes and references==
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| <references />
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| ==External links==
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| * {{MathWorld|title=Heegner Number|urlname=HeegnerNumber}}
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| * {{SloanesRef |sequencenumber=A003173|name=Heegner numbers: imaginary quadratic fields with unique factorization}}
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| * [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.
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| * {{cite web|last=Clark|first=Alex|title=163 and Ramanujan Constant|url=http://www.numberphile.com/videos/163.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| [[Category:Algebraic number theory]]
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