On the SEL Egyptian fraction expansion for real numbers

Mayurachat Janthawee; Narakorn R. Kanasri; Mayurachat Janthawee; Narakorn R. Kanasri

doi:10.3934/math.2022827

AIMS Mathematics

2022, Volume 7, Issue 8: 15094-15106. doi: 10.3934/math.2022827

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Research article

On the SEL Egyptian fraction expansion for real numbers

Mayurachat Janthawee ,
Narakorn R. Kanasri ^,

Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 10 February 2022 Revised: 09 June 2022 Accepted: 14 June 2022 Published: 15 June 2022
MSC : 11A67

In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.
- SEL Egyptian fraction expansion,
- upper bound,
- transcendental number,
- bijection
Citation: Mayurachat Janthawee, Narakorn R. Kanasri. On the SEL Egyptian fraction expansion for real numbers[J]. AIMS Mathematics, 2022, 7(8): 15094-15106. doi: 10.3934/math.2022827

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Abstract

In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.

References

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