On the gap between prime ideals

Tianyu Ni; Tianyu Ni

doi:10.3934/Math.2019.1.79

AIMS Mathematics

2019, Volume 4, Issue 1: 79-85. doi: 10.3934/Math.2019.1.79

Previous Article Next Article

Research article

On the gap between prime ideals

Tianyu Ni ^,

Mathematical College, Sichuan University, Chengdu 610064, China

Received: 18 December 2018 Accepted: 09 January 2019 Published: 16 January 2019
MSC : 11R04, 11D99

We define a gap function to measure the difference of two distinct prime ideals in a given number field. In this paper, we determine all quadratic fields and cyclotomic fields satisfying the condition: there exist two distinct prime ideals whose gap is 1.
- ring of integers,
- quadratic field,
- cyclotomic field,
- prime gap,
- diophantine equations
Citation: Tianyu Ni. On the gap between prime ideals[J]. AIMS Mathematics, 2019, 4(1): 79-85. doi: 10.3934/Math.2019.1.79

Related Papers:

Abstract

We define a gap function to measure the difference of two distinct prime ideals in a given number field. In this paper, we determine all quadratic fields and cyclotomic fields satisfying the condition: there exist two distinct prime ideals whose gap is 1.

References

[1]	J.S. Milne, Algebraic number theory, Lecture Notes, 1998. Available from: http://www.jmilne.org/math/CourseNotes/ant.html.
[2]	J. Neukirch, Algebraic number theory, Berlin: Springer-Verlag, 1999.
[3]	P. Mihäilescu, Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math., 572 (2004), 167-195. doi: 10.1515/crll.2004.048
[4]	T. Metsänkylä, Catalan's conjecture: Another old Diophantine problem solved, B. Am. Math. Soc., 41 (2004), 43-57.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)