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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]].
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==Motivation==
 
A standard way of writing real numbers is by their [[decimal representation]], such as:
:<math>x=a_0.a_1a_2a_3\ldots\ </math>
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>&hellip; are integers between 0 and 9. Given this representation the number ''x'' is equal to
:<math>x=\sum_{n=0}^\infty \frac{a_n}{10^n}.</math>
 
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''&nbsp;&ge;&nbsp;''N'' it is the case that ''a''<sub>''n''+''p''</sub>&nbsp;=&nbsp;''a''<sub>''n''</sub>.
 
Another way of expressing numbers is to write them as [[continued fraction]]s, as in:
:<math>x=[a_0;a_1,a_2,a_3,\ldots],\ </math>
where ''a''<sub>0</sub> is an integer and ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>&hellip; are natural numbers.  From this representation we can recover ''x'' since
:<math>x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \ddots}}}.</math>
 
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''&nbsp;&ge;&nbsp;''N'' we have ''a''<sub>''n''+''p''</sub>&nbsp;=&nbsp;''a''<sub>''n''</sub>), if and only if ''x'' is a [[quadratic irrational]].
 
==Hermite's question==
 
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.
 
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&ndash;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&ndash;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''?
 
==Approaches==
 
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&ndash;64.</ref>  He hoped to show that the sequence attached to (''x'',&nbsp;''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.
 
Rather than generalising continued fractions, another approach to the problem is to generalise [[Minkowski's question mark function]].  This function ?&nbsp;:&nbsp;[0,&nbsp;1]&nbsp;&rarr;&nbsp;[0,&nbsp;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,&nbsp;1]&nbsp;&times;&nbsp;[0,&nbsp;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.&nbsp;1, pp.&nbsp;105&ndash;134.</ref>
 
==References==
{{Reflist}}
 
[[Category:Continued fractions]]
[[Category:Algebraic number theory]]
[[Category:Unsolved problems in mathematics]]

Revision as of 09:30, 7 February 2014

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