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| | Hello! <br>My name is Marjorie and I'm a 25 years old boy from Norway.<br><br>Check out my page [http://makemoney4surveys.com/ how to make money online] |
| In [[complex analysis]], '''de Branges's theorem''', or the '''Bieberbach conjecture''', is a theorem that gives a [[necessary condition]] on a [[holomorphic function]] in order for it to map the [[unit disc|open unit disk]] of the [[complex plane]] [[injective]]ly to the complex plane. It was posed by {{harvs|txt|first=Ludwig |last=Bieberbach|authorlink=Ludwig Bieberbach|year=1916}} and finally proven by {{harvs|txt|authorlink=Louis de Branges de Bourcia|first=Louis |last=de Branges|year=1985}}.
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| The statement concerns the [[Taylor series|Taylor coefficient]]s ''a<sub>n</sub>'' of such a function, normalized as is always possible so that ''a''<sub>0</sub> = 0 and ''a''<sub>1</sub> = 1. That is, we consider a function defined on the open unit disk which is [[holomorphic function|holomorphic]] and injective (''[[Univalent function|univalent]]'') with Taylor series of the form
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| :<math>f(z)=z+\sum_{n\geq 2} a_n z^n</math>
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| such functions are called ''schlicht''. The theorem then states that
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| :<math>\left| a_n \right| \leq n \quad \text{for all }n\geq 2.\,</math>
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| ==Schlicht functions==
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| The normalizations
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| :''a''<sub>0</sub> = 0 and ''a''<sub>1</sub> = 1
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| mean that
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| :''f''(0) = 0 and ''f'' '(0) = 1;
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| this can always be assured by a [[linear fractional transformation]]: starting with an arbitrary injective holomorphic function ''g'' defined on the open unit disk and setting
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| :<math>f(z)=\frac{g(z)-g(0)}{g'(0)}.\,</math>
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| Such functions ''g'' are of interest because they appear in the [[Riemann mapping theorem]].
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| A '''schlicht function''' is defined as an analytic function ''f'' that is one-to-one and satisfies ''f''(0) = 0 and ''f'' '(0) = 1. A family of schlicht functions are the [[rotated Koebe function]]s
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| :<math>f_\alpha(z)=\frac{z}{(1-\alpha z)^2}=\sum_{n=1}^\infty n\alpha^{n-1} z^n</math>
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| with α a complex number of [[absolute value]] 1. If ''f'' is a schlicht function and |''a''<sub>''n''</sub>| = ''n'' for some ''n'' ≥ 2, then ''f'' is a rotated Koebe function.
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| The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function
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| :<math>f(z)=z+z^2 = (z+1/2)^2 - 1/4\;</math>
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| shows: it is holomorphic on the unit disc and satisfies |''a''<sub>''n''</sub>|≤''n'' for all ''n'', but it is not injective since ''f''(−1/2 + ''z'') = ''f''(−1/2 − ''z'').
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| ==History ==
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| A survey of the history is given by [http://kobra.bibliothek.uni-kassel.de/bitstream/urn:nbn:de:hebis:34-200604038936/1/prep0513.pdf Koepf (2007)].
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| {{harvtxt|Bieberbach|1916}} proved |''a''<sub>2</sub>| ≤ 2, and stated the conjecture that |''a''<sub>''n''</sub>| ≤ ''n''. {{harvtxt|Loewner|1917}} and {{harvtxt|Nevanlinna|1921}} independently proved the conjecture for [[Nevanlinna's criterion#Application to Bieberbach conjecture|starlike functions]].
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| Then [[Charles Loewner]] ({{harvtxt|Löwner|1923}}) proved |''a''<sub>3</sub>| ≤ 3, using the [[Löwner equation]]. His work was used by most later attempts, and is also applied in the theory of [[Schramm–Loewner evolution]].
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| {{harvtxt|Littlewood|1925|loc=theorem 20}} proved that |''a''<sub>''n''</sub>| ≤ ''en'' for all ''n'', showing that the Bieberbach conjecture is true up to a factor of ''e'' = 2.718... Several authors later reduced the constant in the inequality below ''e''.
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| If ''f''(''z'') = ''z'' + ... is a schlicht function then φ(''z'') = ''f''(''z''<sup>2</sup>)<sup>1/2</sup> is an odd schlicht function.
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| {{harvs|txt|authorlink=Raymond Paley|last=Paley|author2-link=John Edensor Littlewood|last2=Littlewood|year=1932}} showed that its Taylor coefficients satisfy ''b''<sub>''k''</sub> ≤ 14 for all ''k''. They conjectured that 14 can be replaced by 1 as a natural generalization of the Bieberbach conjecture. The Littlewood–Paley conjecture easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by {{harvtxt|Fekete|Szegö|1933}}, who showed there is an odd schlicht function with ''b''<sub>5</sub> = 1/2 + exp(−2/3) = 1.013..., and that this is the maximum possible value of ''b''<sub>5</sub>. ([[Isaak Moiseevich Milin|Milin]] later showed that 14 can be replaced by 1.14., and Hayman showed that the numbers ''b''<sub>''k''</sub> have a limit less than 1 if φ is not a Koebe function, so Littlewood and Paley's conjecture is true for all but a finite number of coefficients of any function.) A weaker form of Littlewood and Paley's conjecture was found by {{harvtxt|Robertson|1936}}.
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| The '''Robertson conjecture''' states that if
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| :<math>\phi(z) = b_1z+b_3z^3+b_5z^5+\cdots</math>
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| is an odd schlicht function in the unit disk with ''b''<sub>1</sub>=1 then for all positive integers ''n'',
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| :<math>\sum_{k=1}^n|b_{2k+1}|^2\le n.</math>
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| Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for ''n'' = 3. This conjecture introduced the key idea of bounding various quadratic functions of the coefficients rather than the coefficients themselves, which is equivalent to bounding norms of elements in certain Hilbert spaces of schlicht functions.
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| There were several proofs of the Bieberbach conjecture for certain higher values of ''n'', in particular {{harvtxt|Garabedian|Schiffer|1955}} proved |''a''<sub>4</sub>| ≤ 4, {{harvtxt|Ozawa|1969}} and {{harvtxt|Pederson|1968}} proved |''a''<sub>6</sub>| ≤ 6, and {{harvtxt|Pederson|Schiffer|1972}} proved |''a''<sub>5</sub>| ≤ 5.
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| {{harvtxt|Hayman|1955}} proved that the limit of ''a''<sub>''n''</sub>/''n'' exists, and has absolute value less than 1 unless ''f'' is a Koebe function. In particular this showed that for any ''f'' there can be at most a finite number of exceptions to the Bieberbach conjecture.
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| The '''Milin conjecture''' states that for each simple function on the unit disk, and for all positive integers ''n'',
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| :<math>\sum^n_{k=1} (n-k+1)(k|\gamma_k|^2-1/k)\le 0</math>
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| where the '''logarithmic coefficients''' γ<sub>''n''</sub> of ''f'' are given by
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| :<math>\log(f(z)/z)=2 \sum^\infty_{n=1}\gamma_nz^n.</math>
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| {{harvtxt|Milin|1977}} showed using the [[Lebedev–Milin inequality]] that the Milin conjecture (later proved by de Branges) implies the Robertson conjecture and therefore the Bieberbach conjecture.
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| Finally {{harvtxt|De Branges|1985}} proved |''a''<sub>''n''</sub>| ≤ ''n'' for all ''n''.
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| ==De Branges's proof==
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| The proof uses a type of [[Hilbert space]]s of [[entire function]]s. The study of these spaces grew into a sub-field of complex analysis and the spaces come to be called [[de Branges space]]s and the functions [[de Branges function]]s. De Branges proved the stronger Milin conjecture {{harv|Milin|1971}} on logarithmic coefficients. This was already known to imply the Robertson conjecture {{harv|Robertson|1936}} about odd univalent functions, <!--the Rogosinski conjecture {{harv|Rogosinski|1943}} about subordinate functions,--> which in turn was known to imply the Bieberbach conjecture about simple functions {{harv|Bieberbach|1916}}. His proof uses the [[Loewner equation]], the [[Askey–Gasper inequality]] about [[Jacobi polynomial]]s, and the [[Lebedev–Milin inequality]] on exponentiated power series.
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| De Branges reduced the conjecture to some inequalities for Jacobi polynomials, and verified the first few by hand. Walter Gautschi verified more of these inequalities by computer for de Branges (proving the Bieberbach conjecture for the first 30 or so coefficients) and then asked [[Richard Askey]] if he knew of any similar inequalities. Askey pointed out that {{harvtxt|Askey|Gasper|1976}} had proved the necessary inequalities eight years before, which allowed de Branges to complete his proof. The first version was very long and had some minor mistakes which caused some skepticism about it, but these were corrected with the help of members of the [[St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences|Leningrad Department of Steklov Mathematical Institute]] when de Branges visited in 1984.
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| De Branges proved the following result, which for ν = 0 implies the Milin conjecture (and therefore the Bieberbach conjecture).
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| Suppose that ν > −3/2 and σ<sub>''n''</sub> are real numbers for positive integers ''n'' with limit 0 and such that
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| :<math> \rho_n=\frac{\Gamma(2\nu+n+1)}{\Gamma(n+1)}(\sigma_n-\sigma_{n+1}) </math>
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| is non-negative, non-increasing, and has limit 0. Then for all Riemann mapping functions ''F''(''z'') = ''z'' + ... univalent in the unit disk with
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| :<math>\frac{F(z)^\nu-z^\nu} {\nu}= \sum_{n=1}^{\infty} a_nz^{\nu+n}</math>
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| the maximinum value of
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| :<math>\sum_{n=1}^\infty(\nu+n)\sigma_n|a_n|^2</math>
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| is achieved by the Koebe function ''z''/(1 − ''z'')<sup>2</sup>.
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| ==References==
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| *{{Citation | author1-link=Richard Askey | last1=Askey | first1=Richard | last2=Gasper | first2=George | title=Positive Jacobi polynomial sums. II | mr=0430358 | year=1976 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=98 | issue=3 | pages=709–737 | doi=10.2307/2373813 | publisher=American Journal of Mathematics, Vol. 98, No. 3 | jstor=2373813}}
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| *{{Citation | editor1-last=Baernstein | editor1-first=Albert | editor2-last=Drasin | editor2-first=David | editor3-last=Duren | editor3-first=Peter | editor4-last=Marden. | editor4-first=Albert | title=The Bieberbach conjecture | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1521-2 | mr=875226 | year=1986 | volume=21 | pages=xvi+218}}
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| *{{citation|first=L. |last=Bieberbach|title=Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln|journal= Sitzungsber. Preuss. Akad. Wiss. Phys-Math. Kl. |year=1916|pages=940–955}}
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| * {{Citation | last1=Conway | first1=John B. | author1-link=John B. Conway | title=Functions of One Complex Variable II | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94460-9 | year=1995}}
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| *{{Citation | last1=de Branges | first1=Louis | author1-link=Louis de Branges de Bourcia | title=A proof of the Bieberbach conjecture | doi=10.1007/BF02392821 | mr=772434 | year=1985 | journal=[[Acta Mathematica]] | volume=154 | issue=1 | pages=137–152}}
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| *{{Citation | last1=de Branges | first1=Louis | author1-link=Louis de Branges de Bourcia | title=Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=934213 | year=1987 | chapter=Underlying concepts in the proof of the Bieberbach conjecture | pages=25–42}}
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| *{{citation|mr=875226
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| |title=The Bieberbach conjecture
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| |series=Mathematical Surveys and Monographs
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| |volume=21
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| |title=Proceedings of the symposium on the occasion of the proof of the Bieberbach conjecture held at Purdue University, West Lafayette, Ind., March 11—14, 1985
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| |editor1-last=Drasin|editor1-first=David |editor2-last= Duren|editor2-first=Peter |editor3-last=Marden|editor3-first= Albert
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| |publisher=American Mathematical Society
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| |place=Providence, RI
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| |year=1986
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| |pages=xvi+218
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| |isbn= 0-8218-1521-0
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| }}
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| *{{citation|first=M.|last= Fekete |first2= G. |last2=Szegö
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| |title= Eine Bemerkung Über Ungerade Schlichte Funktionen
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| |journal= J. London Math. Soc.|year= 1933 |pages= 85–89| doi=10.1112/jlms/s1-8.2.85|volume=s1-8|issue=2 }}
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| *{{springer|id=B/b016150|first=E.G.|last= Goluzina|title=Bieberbach conjecture}}
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| *{{Citation | last1=Hayman | first1=W. K. | title=The asymptotic behaviour of p-valent functions | doi=10.1112/plms/s3-5.3.257 | mr=0071536 | year=1955 | journal=Proceedings of the London Mathematical Society. Third Series | volume=5 | pages=257–284 | issue=3}}
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| * Koepf, Wolfram (2007), ''[http://kobra.bibliothek.uni-kassel.de/bitstream/urn:nbn:de:hebis:34-200604038936/1/prep0513.pdf Bieberbach’s Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality]''
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| *{{Citation | last1=Korevaar | first1=Jacob | title=Ludwig Bieberbach's conjecture and its proof by Louis de Branges | mr=856290 | year=1986 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=93 | issue=7 | pages=505–514 | doi=10.2307/2323021 | publisher=The American Mathematical Monthly, Vol. 93, No. 7 | jstor=2323021}}
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| *{{citation|first=J. E. |last=Littlewood
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| |title= On Inequalities in the Theory of Functions
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| |journal= Proc. London Math. Soc.|year= 1925 |pages= 481–519| doi=10.1112/plms/s2-23.1.481|volume=s2-23 }}
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| *{{citation|first1=J.E.|last1= Littlewood |first2= E. A. C.|last2= Paley
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| |title= A Proof That An Odd Schlicht Function Has Bounded Coefficients
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| |journal= J. London Math. Soc.|year= 1932 |pages= 167–169| doi=10.1112/jlms/s1-7.3.167|volume=s1-7|issue=3 }}
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| *{{citation|first=C.|last= Loewner|title= Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheitskreises /z/ < 1, die durch Funktionen mit nicht verschwindender Ableitung geliefert werden|journal=Ber. Verh. Sachs. Ges. Wiss. Leipzig|volume= 69|year= 1917|pages= 89–106}}
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| *{{citation|first=C. |last=Loewner|title=Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I|journal=Math. Ann. |volume=89|year=1923|pages= 103–121|id= [http://www.emis.de/cgi-bin/JFM-item?49.0714.01 JFM 49.0714.01]|doi=10.1007/BF01448091}}
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| *{{Citation | last1=Milin | first1=I. M. | title=Univalent functions and orthonormal systems | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0369684 | year=1977}} (Translation of the 1971 Russian edition)
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| *{{citation|last=Nevanlinna|first= R.|title=Über die konforme Abbildung von Sterngebieten|journal=Ofvers. Finska Vet. Soc. Forh. |volume=53 |year=1921|pages=1–21}}
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| *{{Citation | last1=Robertson | first1=M. S. | title=A remark on the odd schlicht functions | url=http://www.ams.org/bull/1936-42-06/S0002-9904-1936-06300-7/ | doi=10.1090/S0002-9904-1936-06300-7 | year=1936 | journal=[[Bulletin of the American Mathematical Society]] | volume=42 | pages=366–370 | issue=6}}
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| [[Category:Theorems in complex analysis]]
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| [[Category:Conjectures]]
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