Editing Heegner number

{{Short description|Concept in algebraic number theory}}
In [[number theory]], a '''Heegner number''' (as termed by [[John Horton Conway|Conway]] and [[Richard K. Guy|Guy]]) is a [[Square-free integer|square-free positive integer]] ''d'' such that the imaginary [[quadratic field]] <math>\Q\left[\sqrt{-d}\right]</math> has [[ideal class group|class number]] 1. Equivalently, the [[ring of integers|ring of algebraic integers]] of <math>\Q\left[\sqrt{-d}\right]</math> has [[unique factorization]].<ref>{{cite book
  | last = Conway
  | first = John Horton
  | authorlink = John Horton Conway
  | author2 = Guy, Richard K.
  | title = The Book of Numbers
  | publisher = Springer
  | year = 1996
  | page = [https://archive.org/details/bookofnumbers0000conw/page/224 224]
  | isbn = 0-387-97993-X
  | url = https://archive.org/details/bookofnumbers0000conw/page/224
  }}
</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 (Baker–)[[Stark–Heegner theorem]] there are precisely nine Heegner numbers:
{{block indent|left=1.6|1, 2, 3, 7, 11, 19, 43, 67, and 163. {{OEIS|A003173}}}}
This result was conjectured by [[Carl Friedrich Gauss|Gauss]] and proved up to minor flaws by [[Kurt Heegner]] in 1952. [[Alan Baker (mathematician)|Alan Baker]] and [[Harold Stark]] independently proved the result in 1966, and Stark further indicated that the gap in Heegner's proof was minor.<ref>{{citation|last=Stark|first=H. M.|authorlink=Harold Stark|year=1969|url=http://deepblue.lib.umich.edu/bitstream/2027.42/33039/1/0000425.pdf|title=On the gap in the theorem of Heegner|journal=[[Journal of Number Theory]]|volume=1|issue=1|pages=16&ndash;27|doi=10.1016/0022-314X(69)90023-7|bibcode=1969JNT.....1...16S|hdl=2027.42/33039|hdl-access=free}}</ref>

==Euler's prime-generating polynomial==
Euler's [[Formula for primes#Prime formulas and polynomial functions|prime-generating polynomial]]
<math display=block>n^2 + n + 41,</math>
which gives (distinct) primes for ''n''&nbsp;=&nbsp;0,&nbsp;...,&nbsp;39, is related to the Heegner number 163&nbsp;=&nbsp;4&nbsp;·&nbsp;41&nbsp;−&nbsp;1.

[[George Yuri Rainich|Rabinowitsch]]<ref>[[George Yuri Rainich|Rabinovitch, Georg]] [https://babel.hathitrust.org/cgi/pt?id=miun.aag4063.0001.001;view=1up;seq=420 "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern."] Proc. Fifth Internat. Congress Math. ( Cambridge) 1, 418–421, 1913.</ref> proved that
<math display=block>n^2 + n + p</math>
gives primes for <math>n=0,\dots,p-2</math> if and only if this quadratic's [[discriminant]] <math>1-4p</math> is the negative of 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 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 [[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]]'':<ref>[http://mathworld.wolfram.com/RamanujanConstant.html Ramanujan Constant – from Wolfram MathWorld<!-- Bot-generated title -->]</ref>
<math display=block>e^{\pi \sqrt{163}} = 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots\approx 640\,320^3+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
  | page = 72
  | 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 | doi = 10.1038/scientificamerican0475-126
 | bibcode = 1975SciAm.232d.126G
 }}
</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. In this wise it has as a spurious provenance as the [[Feynman point]].

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

===Detail===
In what follows, j(z) denotes the [[j-invariant]] of the [[complex number]] z. Briefly, <math>\textstyle j\left(\frac{1+\sqrt{-d}}{2}\right)</math> is an integer for&nbsp;''d'' a Heegner number, and
<math display=block>e^{\pi \sqrt{d}} \approx -j\left(\frac{1+\sqrt{-d}}{2}\right) + 744</math>
via the ''q''-expansion.

If <math>\tau</math> is a quadratic irrational, then its ''j''-invariant <math>j(\tau)</math> is an [[algebraic integer]] of degree <math>\left|\mathrm{Cl}\bigl(\mathbf{Q}(\tau)\bigr)\right|</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=e^{2 \pi i \tau}</math>, begins as:
<math display=block>j(\tau) = \frac{1}{q} + 744 + 196\,884 q + \cdots.</math>

The coefficients <math>c_n</math> asymptotically grow as
<math display=block>\ln(c_n) \sim 4\pi \sqrt{n} + O\bigl(\ln(n)\bigr),</math>
and the low order coefficients grow more slowly than <math>200\,000^n</math>, so for <math>\textstyle q \ll \frac{1}{200\,000}</math>, ''j'' is very well approximated by its first two terms.  Setting <math>\textstyle\tau = \frac{1+\sqrt{-163}}{2}</math> yields
<math display=block> q=-e^{-\pi \sqrt{163}} \quad\therefore\quad \frac{1}{q}=-e^{\pi \sqrt{163}}. </math>
Now
<math display=block>j\left(\frac{1+\sqrt{-163}}{2}\right)=\left(-640\,320\right)^3,</math>
so,
<math display=block>\left(-640\,320\right)^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right).</math>
Or,
<math display=block>e^{\pi \sqrt{163}}=640\,320^3+744+O\left(e^{-\pi \sqrt{163}}\right)</math>
where the linear term of the error is,
<math display=block>\frac{-196\,884}{e^{\pi \sqrt{163}}} \approx \frac{-196\,884}{640\,320^3+744}
\approx -0.000\,000\,000\,000\,75</math>
explaining why <math>e^{\pi \sqrt{163}}</math> is within approximately the above of being an integer.

==Pi formulas==

The [[Chudnovsky brothers]] found in 1987 that
<math display=block>\frac{1}{\pi} = \frac{12}{640\,320^\frac32} \sum_{k=0}^\infty \frac{(6k)! (163 \cdot 3\,344\,418k + 13\,591\,409)}{(3k)!(k!)^3 (-640\,320)^{3k}},</math>
a proof of which uses the fact that
<math display=block>j\left(\frac{1+\sqrt{-163}}{2}\right) = -640\,320^3.</math>
For similar formulas, see the [[Ramanujan–Sato series]].

==Other Heegner numbers==
For the four largest Heegner numbers, the approximations one obtains<ref>These can be checked by computing
<math display=block>\sqrt[3]{e^{\pi\sqrt{d}}-744}</math>
on a calculator, and
<math display=block>\frac{196\,884}{e^{\pi\sqrt{d}}}</math>
for the linear term of the error.</ref> are as follows.
<math display=block>\begin{align}
e^{\pi \sqrt{19}}  &\approx \phantom{000\,0}96^3+744-0.22\\
e^{\pi \sqrt{43}}  &\approx \phantom{000\,}960^3+744-0.000\,22\\
e^{\pi \sqrt{67}}  &\approx \phantom{00}5\,280^3+744-0.000\,0013\\
e^{\pi \sqrt{163}} &\approx 640\,320^3+744-0.000\,000\,000\,000\,75
\end{align}
</math>

Alternatively,<ref>{{Cite web|url=http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en|title=More on e^(pi*SQRT(163))|access-date=2008-04-19|archive-date=2009-08-11|archive-url=https://web.archive.org/web/20090811214935/http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en|url-status=dead}}</ref>
<math display=block>\begin{align}
e^{\pi \sqrt{19}}  &\approx 12^3\left(3^2-1\right)^3\phantom{00}+744-0.22\\
e^{\pi \sqrt{43}}  &\approx 12^3\left(9^2-1\right)^3\phantom{00}+744-0.000\,22\\
e^{\pi \sqrt{67}}  &\approx 12^3\left(21^2-1\right)^3\phantom{0}+744-0.000\,0013\\
e^{\pi \sqrt{163}} &\approx 12^3\left(231^2-1\right)^3+744-0.000\,000\,000\,000\,75
\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 form
:<math display=block>12^3\left(n^2-1\right)^3=\left(2^2\cdot 3 \cdot (n-1) \cdot (n+1)\right)^3,</math>
and factor as,
<math display=block>\begin{align}
j\left(\frac{1+\sqrt{-19}}{2}\right) &= \phantom{000\,0}-96^3 = -\left(2^5 \cdot 3\right)^3\\
j\left(\frac{1+\sqrt{-43}}{2}\right) &= \phantom{000\,}-960^3 = -\left(2^6 \cdot 3 \cdot 5\right)^3\\
j\left(\frac{1+\sqrt{-67}}{2}\right) &= \phantom{00}-5\,280^3 = -\left(2^5 \cdot 3 \cdot 5 \cdot 11\right)^3\\
j\left(\frac{1+\sqrt{-163}}{2}\right)&= -640\,320^3 = -\left(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29\right)^3.
\end{align}
</math>

These [[transcendental numbers]], in addition to being closely approximated by integers (which are simply [[algebraic numbers]] of degree&nbsp;1), can 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 display=block>\begin{align}
e^{\pi \sqrt{19}}  &\approx x^{24}-24.000\,31 ; & x^3-2x-2&=0\\
e^{\pi \sqrt{43}}  &\approx x^{24}-24.000\,000\,31 ; & x^3-2x^2-2&=0\\
e^{\pi \sqrt{67}}  &\approx x^{24}-24.000\,000\,0019 ; & x^3-2x^2-2x-2&=0\\
e^{\pi \sqrt{163}} &\approx x^{24}-24.000\,000\,000\,000\,0011  ; &\quad 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. 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 display=block>\begin{align}
e^{\pi \sqrt{19}}  &\approx 3^5 \left(3-\sqrt{2\left(1- \tfrac{96}{24}+1\sqrt{3\cdot19}\right)} \right)^{-2}-12.000\,06\dots\\
e^{\pi \sqrt{43}}  &\approx 3^5 \left(9-\sqrt{2\left(1- \tfrac{960}{24}+7\sqrt{3\cdot43}\right)} \right)^{-2}-12.000\,000\,061\dots\\
e^{\pi \sqrt{67}}  &\approx 3^5 \left(21-\sqrt{2\left(1- \tfrac{5\,280}{24} +31\sqrt{3\cdot67}\right)} \right)^{-2}-12.000\,000\,000\,36\dots\\
e^{\pi \sqrt{163}}  &\approx 3^5 \left(231-\sqrt{2\left(1- \tfrac{640\,320}{24}+2\,413\sqrt{3\cdot163}\right)} \right)^{-2}-12.000\,000\,000\,000\,000\,21\dots
\end{align}
</math>

If <math>x</math> denotes the expression within the parenthesis (e.g. <math>x=3-\sqrt{2\left(1- \tfrac{96}{24}+1\sqrt{3\cdot19}\right)}</math>), it satisfies respectively the [[quartic equation]]s 
<math display=block>\begin{align}
x^4 -\phantom{00} 4\cdot 3 x^3 + \phantom{000\,0}\tfrac23( 96  +3) x^2 - \phantom{000\,000}\tfrac23\cdot3(96-6)x - 3&=0\\
x^4 -\phantom{00} 4\cdot 9x^3 + \phantom{000\,}\tfrac23( 960  +3) x^2 - \phantom{000\,00}\tfrac23\cdot9(960-6)x - 3&=0\\
x^4 -\phantom{0} 4\cdot 21x^3 + \phantom{00}\tfrac23( 5\,280  +3) x^2 - \phantom{000}\tfrac23\cdot21(5\,280-6)x - 3&=0\\
x^4 - 4\cdot 231x^3 + \tfrac23( 640\,320  +3) x^2 - \tfrac23\cdot231(640\,320-6)x - 3&=0\\
\end{align}
</math>

Note the reappearance of the integers <math>n = 3, 9, 21, 231</math> as well as the fact that
<math display=block>\begin{align}
2^6 \cdot 3\left(-\left(1- \tfrac{96}{24}\right)^2+ 1^2 \cdot3\cdot 19 \right) &= 96^2\\
2^6 \cdot 3\left(-\left(1- \tfrac{960}{24}\right)^2+ 7^2\cdot3 \cdot 43 \right) &= 960^2\\
2^6 \cdot 3\left(-\left(1- \tfrac{5\,280}{24}\right)^2+ 31^2 \cdot 3\cdot67 \right) &= 5\,280^2\\
2^6 \cdot 3\left(-\left(1- \tfrac{640\,320}{24}\right)^2+ 2413^2\cdot 3 \cdot163 \right) &= 640\,320^2
\end{align}
</math>
which, with the appropriate fractional power, are precisely the ''j''-invariants.

Similarly for algebraic numbers of degree&nbsp;6,
<math display=block>\begin{align}
e^{\pi \sqrt{19}}  &\approx \left(5x\right)^3-6.000\,010\dots\\
e^{\pi \sqrt{43}}  &\approx \left(5x\right)^3-6.000\,000\,010\dots\\
e^{\pi \sqrt{67}}  &\approx \left(5x\right)^3-6.000\,000\,000\,061\dots\\
e^{\pi \sqrt{163}} &\approx \left(5x\right)^3-6.000\,000\,000\,000\,000\,034\dots
\end{align}
</math>

where the ''x''s are given respectively by the appropriate root of the [[sextic equation]]s,
<math display=block>\begin{align}
5x^6-\phantom{000\,0}96x^5-10x^3+1&=0\\
5x^6-\phantom{000\,}960x^5-10x^3+1&=0\\
5x^6-\phantom{00}5\,280x^5-10x^3+1&=0\\
5x^6-640\,320x^5-10x^3+1&=0
\end{align}
</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>\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>\textstyle \tau = \frac{1+\sqrt{-163}}{2}</math>, then,
<math display=block>\begin{align}
e^{\pi \sqrt{163}} &= \left( \frac{e^\frac{\pi i}{24} \eta(\tau)}{\eta(2\tau)} \right)^{24}-24.000\,000\,000\,000\,001\,05\dots\\
e^{\pi \sqrt{163}} &= \left( \frac{e^\frac{\pi i}{12} \eta(\tau)}{\eta(3\tau)} \right)^{12}-12.000\,000\,000\,000\,000\,21\dots\\
e^{\pi \sqrt{163}} &= \left( \frac{e^\frac{\pi i}{6} \eta(\tau)}{\eta(5\tau)} \right)^{6}-6.000\,000\,000\,000\,000\,034\dots
\end{align}
</math>

where the eta quotients are the algebraic numbers given above.

==Class 2 numbers==
The three numbers 88, 148, 232, for which the imaginary [[quadratic field]] <math>\Q\left[\sqrt{-d}\right]</math> has [[ideal class group|class number]] 2, are not Heegner numbers but have certain similar properties in terms of [[almost integer]]s. For instance,<ref>{{Cite web|author=Titus Piezas|url=https://www.oocities.org/titus_piezas/Ramanujan_a.pdf|title=Ramanujan’s Constant e^(pv163) And Its Cousins}}</ref> 
<math display=block>\begin{align}
e^{\pi \sqrt{88}} +8\,744 &\approx \phantom{00\,00}2\,508\,952^2-0.077\dots\\
e^{\pi \sqrt{148}} +8\,744 &\approx \phantom{00\,}199\,148\,648^2-0.000\,97\dots\\
e^{\pi \sqrt{232}} +8\,744 &\approx 24\,591\,257\,752^2-0.000\,0078\dots\\
\end{align}
</math>
and
<math display=block>\begin{align}
e^{\pi \sqrt{22}} -24  &\approx \phantom{00}\left(6+4\sqrt{2}\right)^{6} +0.000\,11\dots\\
e^{\pi \sqrt{37}} +24  &\approx \left(12+ 2 \sqrt{37}\right)^6 -0.000\,0014\dots\\
e^{\pi \sqrt{58}} -24  &\approx \left(27 + 5 \sqrt{29}\right)^6 -0.000\,000\,0011\dots\\
\end{align}
</math>

==Consecutive primes==
Given an odd prime&nbsp;''p'', if one computes <math>k^2 \mod p</math> for <math>\textstyle k=0,1,\dots,\frac{p-1}{2}</math> (this is sufficient because <math>\left(p-k\right)^2\equiv k^2\mod p</math>), one gets consecutive composites, followed by consecutive primes, if and only if ''p'' is a Heegner number.<ref>{{Cite web|url=http://www.mathpages.com/home/kmath263.htm|title=Simple Complex Quadratic Fields}}</ref>

For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by [[Richard Mollin]].<ref>{{cite journal|author=Mollin, R. A.|title=Quadratic polynomials producing consecutive, distinct primes and class groups of complex quadratic fields|journal=Acta Arithmetica|volume=74|year=1996|pages=17–30|doi=10.4064/aa-74-1-17-30|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa74/aa7412.pdf}}</ref>

==Notes and references==
{{reflist}}

==External links==
* {{MathWorld|title=Heegner Number|urlname=HeegnerNumber}}
* {{OEIS el|sequencenumber=A003173|name=Heegner numbers: imaginary quadratic fields with unique factorization|formalname=Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1)}}
* [https://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]]|access-date=2013-04-02|archive-url=https://web.archive.org/web/20130516045906/http://www.numberphile.com/videos/163.html|archive-date=2013-05-16|url-status=dead}}

[[Category:Algebraic number theory]]