Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.
Motivation A standard way of writing real numbers is by their decimal representation, such as:
x = a 0 . a 1 a 2 a 3 … {\displaystyle x=a_{0}.a_{1}a_{2}a_{3}\ldots \ }
where a0 is an integer, the integer part of x, and a1, a2, a3, ... are integers between 0 and 9. Given this representation the number x is equal to
x = ∑ n = 0 ∞ a n 10 n . {\displaystyle x=\sum _{n=0}^{\infty }{\frac {a_{n}}{10^{n}}}.}
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 an+p = an. Another way of expressing numbers is to write them as simple continued fractions, as in:
x = [ a 0 ; a 1 , a 2 , a 3 , … ] , {\displaystyle x=[a_{0};a_{1},a_{2},a_{3},\ldots ],\ }
where a0 is an integer and a1, a2, a3... are natural numbers. From this representation we can recover x since
x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + ⋱ . {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\ddots }}}}}}.}
If x is a rational number then the sequence (an) terminates after finitely many terms. On the other hand, Euler proved that irrational numbers require an infinite sequence to express them as continued fractions. Moreover, this sequence is eventually periodic (again, so that there are natural numbers N and p such that for every n ≥ N we have an+p = an), if and only if x is a quadratic irrational.
Hermite's question Rational numbers are algebraic numbers 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 (an) 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? 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?
… excerpt ends here. Continue reading the full article.
