# Algebraic Numbers as Product of Powers of Transcendental Numbers

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Proof of Theorem 2

#### 2.1. Auxiliary Results

**Lemma**

**1**

**.**If ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}$ are algebraic numbers other than 0 or 1, ${\beta}_{1},{\beta}_{2},\dots ,{\beta}_{n}$ are algebraic with $1,{\beta}_{1},{\beta}_{2},\dots ,{\beta}_{n}$ linearly independent over $\mathbb{Q}$, then ${\alpha}_{1}^{{\beta}_{1}}{\alpha}_{2}^{{\beta}_{2}}\cdots {\alpha}_{n}^{{\beta}_{n}}$ is transcendental.

**Lemma**

**2**

**.**Any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental.

**Lemma**

**3.**

**Proof of Lemma**

**3.**

#### 2.2. The Proof

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Value of x | Class of Numbers | Value of y | Class of Numbers | Power ${\mathit{x}}^{\mathit{y}}$ | Class of Numbers |
---|---|---|---|---|---|

2 | algebraic | $log3/log2$ | transcendental | 3 | algebraic |

2 | algebraic | $ilog3/log2$ | transcendental | ${3}^{i}$ | transcendental |

${e}^{i}$ | transcendental | $\pi $ | transcendental | −1 | algebraic |

e | transcendental | $\pi $ | transcendental | ${e}^{\pi}$ | transcendental |

${2}^{\sqrt{2}}$ | transcendental | $\sqrt{2}$ | algebraic | 4 | algebraic |

${2}^{\sqrt{2}}$ | transcendental | $i\sqrt{2}$ | algebraic | ${4}^{i}$ | transcendental |

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**MDPI and ACS Style**

Trojovský, P.
Algebraic Numbers as Product of Powers of Transcendental Numbers. *Symmetry* **2019**, *11*, 887.
https://doi.org/10.3390/sym11070887

**AMA Style**

Trojovský P.
Algebraic Numbers as Product of Powers of Transcendental Numbers. *Symmetry*. 2019; 11(7):887.
https://doi.org/10.3390/sym11070887

**Chicago/Turabian Style**

Trojovský, Pavel.
2019. "Algebraic Numbers as Product of Powers of Transcendental Numbers" *Symmetry* 11, no. 7: 887.
https://doi.org/10.3390/sym11070887