In mathematics, a sequence of positive integers ''a''_''n'' is called an irrationality sequence if it has the property that for every sequence ''x''_''n'' of positive integers, the sum of the series :

\sum_^\infty \frac

exists (that is, it converges) and is an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

.. 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,

2^

, form an irrationality sequence. However, although

Sylvester's sequence In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are :2, 3, 7, 43, 1807, 3263443, 10650056950807, 11342371305542184 ...

: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

x_n=1

for all

n

gives :

\frac+\frac+\frac+\frac+\cdots = 1,

a series converging to a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

. Likewise, the factorials,

n!

, do not form an irrationality sequence because the sequence given by

x_n=n+2

for all

n

leads to a series with a rational sum, :

\sum_^\frac=\frac+\frac+\frac+\frac+\frac+\cdots=1.

Growth rate

For any sequence ''a''_''n'' to be an irrationality sequence, it must grow at a rate such that :

\limsup_ \frac \geq \log 2

. 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 :

\lim_ a_n^=\infty.

However, it is not known whether there exists such a sequence in which the

greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

of each pair of terms is 1 (unlike the powers of powers of two) and for which :

\lim_ a_n^<\infty.

Related properties

Analogously to irrationality sequences, has defined a transcendental sequence to be an integer sequence ''a''_''n'' such that, for every sequence ''x''_''n'' of positive integers, the sum of the series :

\sum_^\infty \frac

exists and is a

transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...

References

{{reflist Integer sequences Irrational numbers Number theory