## Erdős’ method for determining the irrationality of products.(English)Zbl 1242.11050

The main result of the paper is the following: Let $$\varepsilon$$ be a positive real number. Let $$\{a_n\}^\infty_{n=1}$$ and $$\{b_n\}^\infty_{n=1}$$ be two sequences of positive integers. Assume $$\{a_n\}^\infty_{n=1}$$ is nondecreasing and $\limsup_{n\to\infty} a^{1/2^n}_n= \infty.$ Assume that for all sufficiently large $$n$$ $n^{1+\varepsilon}\leq a_n\quad\text{and}\quad b_n\leq a^{1/\log^{1+\varepsilon}\log a_n}.$ Then the number $$x=\prod^\infty_{n=1} (1+(b_n/a_n))$$ is irrational.
For the proof assume that $$x={p\over q}$$, $$(p, q)\in\mathbb{Z}^+\times \mathbb{Z}^+$$. Than for each $$(P, Q)\in\mathbb{Z}^+\times \mathbb{Z}^+$$ one has $\Biggl| qQ\Biggl(x-{P\over Q}\Biggr)\Biggr|= |pQ- Pq|$ and now one can derive in several steps and with some ability the contradiction that there exists $a(P, Q)\quad\text{with}\quad 0< |pQ- Pq|< 1.$ A direct application is the following result:
Let $$\{a_n\}^\infty_{n=1}$$ be an increasing sequence of positive integers such that $$\lim_{n\to\infty} a^{1/2^n}_n= \infty$$.
Then the number $$\prod^\infty_{n=1} (1+ 1/a_n)^{n\to\infty}$$ is irrational.
In the introduction some links are given to results connected with similar infinite products and some hints are given to some papers of Erdős.

### MSC:

 11J72 Irrationality; linear independence over a field

### Keywords:

irrationality; infinite products
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### References:

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