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
exists and is an irrational number. 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".

Examples

The powers of two whose exponents are powers of two,, form an irrationality sequence. However, although Sylvester's sequence
also grows doubly exponentially, it does not form an irrationality sequence. For, letting gives
a series converging to a rational number. Likewise, the factorials,, do not form an irrationality sequence because the sequence leads to a series with a rational sum,

Growth rate

For any sequence an to be an irrationality sequence, it must grow at a rate such that
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.
Every irrationality sequence must grow quickly enough that
However, it is not known whether there exists such a sequence in which the greatest common divisor of each pair of terms is 1 and for which

Related properties

Analogously to irrationality sequences,
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
exists and is a transcendental number.