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| '''Hermite's problem''' is an open problem in [[mathematics]] posed by [[Charles Hermite]] in 1848. He asked for a way of expressing [[real number]]s as sequences of [[natural number]]s, such that the sequence is eventually periodic precisely when the original number is a cubic [[Irrational number|irrational]].
| | Частное предприятие «Илигран»<br>220073, [http://iligran.by/%d0%b0%d1%80%d0%b5%d0%bd%d0%b4%d0%b0-%d0%b1%d0%b0%d1%88%d0%b5%d0%bd%d0%bd%d1%8b%d1%85-%d0%ba%d1%80%d0%b0%d0%bd%d0%be%d0%b2/ башенный кран Минск] г. Минск, ул. Каль[http://iligran.by/%d0%bd%d0%b0%d1%88%d0%b0-%d1%82%d0%b5%d1%85%d0%bd%d0%b8%d0%ba%d0%b0-%d0%b2-%d0%b0%d1%80%d0%b5%d0%bd%d0%b4%d1%83/ башенный кран в аренду Минск]арийская, дом 25, офис 424<br>Телефоны:<br><br> |
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| ==Motivation==
| | +375 44 545-67-00<br><br>+375 29 379-91-88<br>+375 17 204 42 28 (факс)<br>+375 17 204 42 26 (факс)<br>+375 17 204 01 72<br>Email: 2044228@mail.ru<br><br>http://iligran.by |
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| A standard way of writing real numbers is by their [[decimal representation]], such as:
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| :<math>x=a_0.a_1a_2a_3\ldots\ </math>
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| where ''a''<sub>0</sub> is an integer, the [[Floor and ceiling functions|integer part]] of ''x'', and ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>… are integers between 0 and 9. Given this representation the number ''x'' is equal to
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| :<math>x=\sum_{n=0}^\infty \frac{a_n}{10^n}.</math>
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| The real number ''x'' is a [[rational number]] only if its decimal expansion is eventually periodic, that is if there are natural numbers ''N'' and ''p'' such that for every ''n'' ≥ ''N'' it is the case that ''a''<sub>''n''+''p''</sub> = ''a''<sub>''n''</sub>.
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| Another way of expressing numbers is to write them as [[continued fraction]]s, as in:
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| :<math>x=[a_0;a_1,a_2,a_3,\ldots],\ </math>
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| where ''a''<sub>0</sub> is an integer and ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>… are natural numbers. From this representation we can recover ''x'' since
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| :<math>x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \ddots}}}.</math>
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| If ''x'' is a rational number then the sequence (''a''<sub>''n''</sub>) terminates after finitely many terms. On the other hand, Euler proved that irrational numbers require an infinite sequence to express them as continued fractions.<ref>{{cite web | title=E101 – Introductio in analysin infinitorum, volume 1|url=http://math.dartmouth.edu/~euler/pages/E101.html| accessdate=2008-03-16}}</ref> Moreover, this sequence is eventually periodic (again, so that there are natural numbers ''N'' and ''p'' such that for every ''n'' ≥ ''N'' we have ''a''<sub>''n''+''p''</sub> = ''a''<sub>''n''</sub>), if and only if ''x'' is a [[quadratic irrational]].
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| ==Hermite's question==
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| Rational numbers are [[algebraic number]]s that satisfy a polynomial of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these sets of numbers we have a way to construct a sequence of natural numbers (''a''<sub>''n''</sub>) with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and only if the sequence is eventually periodic.
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| In 1848 Charles Hermite wrote a letter to [[Carl Gustav Jacob Jacobi]] asking if this situation could be generalised, that is can one assign a sequence of natural numbers to each real number ''x'' such that the sequence is eventually periodic precisely when ''x'' is a cubic irrational, that is an algebraic number of degree 3?<ref>Émile Picard, ''L'œuvre scientifique de Charles Hermite'', Ann. Sci. École Norm. Sup. '''3''' 18 (1901), pp.9–34.</ref><ref>''Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres. (Continuation).'', Journal für die reine und angewandte Mathematik '''40''' (1850), pp.279–315, {{doi|10.1515/crll.1850.40.279}}</ref> Or, more generally, for each natural number ''d'' is there a way of assigning a sequence of natural numbers to each real number ''x'' that can pick out when ''x'' is algebraic of degree ''d''?
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| ==Approaches==
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| Sequences that attempt to solve Hermite's problem are often called [[Generalized continued fraction#Higher dimensions|multidimensional continued fractions]]. Jacobi himself came up with an early example, finding a sequence corresponding to each pair of real numbers (''x'',''y'') that acted as a higher dimensional analogue of continued fractions.<ref>C. G. J. Jacobi, ''Allgemeine Theorie der kettenbruchänlichen Algorithmen, in welche jede Zahl aus ''drei'' vorhergehenden gebildet wird'' (English: ''General theory of continued-fraction-like algorithms in which each number is formed from three previous ones''), Journal für die reine und angewandte Mathematik '''69''' (1868), pp.29–64.</ref> He hoped to show that the sequence attached to (''x'', ''y'') was eventually periodic if and only if both ''x'' and ''y'' belonged to a [[Cubic field|cubic number field]], but was unable to do so and whether this is the case remains unsolved.
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| Rather than generalising continued fractions, another approach to the problem is to generalise [[Minkowski's question mark function]]. This function ? : [0, 1] → [0, 1] also picks out quadratic irrational numbers since ?(''x'') is rational if and only if ''x'' is either rational or a quadratic irrational number, and moreover ''x'' is rational if and only if ?(''x'') is a [[dyadic rational]], thus ''x'' is a quadratic irrational precisely when ?(''x'') is a non-dyadic rational number. Various generalisations of this function to either the unit square [0, 1] × [0, 1] or the two-dimensional [[simplex]] have been made, though none has yet solved Hermite's problem.<ref>L. Kollros, ''Un Algorithme pour L'Aproximation simultanée de Deux Granduers'', Inaugural-Dissertation, Universität Zürich, 1905.</ref><ref>Olga R. Beaver, Thomas Garrity, ''A two-dimensional Minkowski ?(x) function'', J. Number Theory '''107''' (2004), no. 1, pp. 105–134.</ref>
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| ==References==
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| {{Reflist}}
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| [[Category:Continued fractions]]
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| [[Category:Algebraic number theory]]
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| [[Category:Unsolved problems in mathematics]]
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Частное предприятие «Илигран»
220073, башенный кран Минск г. Минск, ул. Кальбашенный кран в аренду Минскарийская, дом 25, офис 424
Телефоны:
+375 44 545-67-00
+375 29 379-91-88
+375 17 204 42 28 (факс)
+375 17 204 42 26 (факс)
+375 17 204 01 72
Email: 2044228@mail.ru
http://iligran.by