Irrationality sequence
In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series
- [math]\displaystyle{ \sum_{n=1}^\infty \frac{1}{a_n x_n} }[/math]
exists (that is, it converges) and is an irrational number.[1][2] The problem of characterizing irrationality sequences was posed by Paul Erdős and Ernst G. Straus, who originally called the property of being an irrationality sequence "Property P".[3]
Examples
The powers of two whose exponents are powers of two, [math]\displaystyle{ 2^{2^n} }[/math], form an irrationality sequence. However, although Sylvester's sequence
- 2, 3, 7, 43, 1807, 3263443, ...
(in which each term is one more than the product of all previous terms) also grows doubly exponentially, it does not form an irrationality sequence. For, letting [math]\displaystyle{ x_n=1 }[/math] for all [math]\displaystyle{ n }[/math] gives
- [math]\displaystyle{ \frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots = 1, }[/math]
a series converging to a rational number. Likewise, the factorials, [math]\displaystyle{ n! }[/math], do not form an irrationality sequence because the sequence given by [math]\displaystyle{ x_n=n+2 }[/math] for all [math]\displaystyle{ n }[/math] leads to a series with a rational sum,
- [math]\displaystyle{ \sum_{n=0}^{\infty}\frac{1}{(n+2)n!}=\frac{1}{2}+\frac{1}{3}+\frac{1}{8}+\frac{1}{30}+\frac{1}{144}+\cdots=1. }[/math][1]
Growth rate
For any sequence an to be an irrationality sequence, it must grow at a rate such that
- [math]\displaystyle{ \limsup_{n\to\infty} \frac{\log\log a_n}{n} \geq \log 2 }[/math].[4]
This includes sequences that grow at a more than doubly exponential rate as well as some doubly exponential sequences that grow more quickly than the powers of powers of two.[1]
Every irrationality sequence must grow quickly enough that
- [math]\displaystyle{ \lim_{n\to\infty} a_n^{1/n}=\infty. }[/math]
However, it is not known whether there exists such a sequence in which the greatest common divisor of each pair of terms is 1 (unlike the powers of powers of two) and for which
- [math]\displaystyle{ \lim_{n\to\infty} a_n^{1/2^n}\lt \infty. }[/math][5]
Related properties
Analogously to irrationality sequences, (Hančl 1996) has defined a transcendental sequence to be an integer sequence an such that, for every sequence xn of positive integers, the sum of the series
- [math]\displaystyle{ \sum_{n=1}^\infty \frac{1}{a_n x_n} }[/math]
exists and is a transcendental number.[6]
References
- ↑ 1.0 1.1 1.2 Guy, Richard K. (2004), "E24 Irrationality sequences", Unsolved problems in number theory (3rd ed.), Springer-Verlag, p. 346, ISBN 0-387-20860-7, https://books.google.com/books?id=1AP2CEGxTkgC&pg=PA346.
- ↑ Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique, 28, Geneva: Université de Genève L'Enseignement Mathématique, 1980, p. 128.
- ↑ "Some problems and results on the irrationality of the sum of infinite series", Journal of Mathematical Sciences 10: 1–7 (1976), 1975, http://www.renyi.hu/~p_erdos/1976-44.pdf.
- ↑ Hančl, Jaroslav (1991), "Expression of real numbers with the help of infinite series", Acta Arithmetica 59 (2): 97–104, doi:10.4064/aa-59-2-97-104
- ↑ "On the irrationality of certain series: problems and results", New advances in transcendence theory (Durham, 1986), Cambridge: Cambridge Univ. Press, 1988, pp. 102–109, http://www.renyi.hu/~p_erdos/1988-22.pdf.
- ↑ Hančl, Jaroslav (1996), "Transcendental sequences", Mathematica Slovaca 46 (2–3): 177–179.
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