k-Diophantine m-tuples in finite fields-Reference-Cited by-同舟云学术

k-Diophantine m-tuples in finite fields

Published:2022-11-16 Issue:04 Volume:19 Page:891-912
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

Hammonds Trajan¹,Kim Seoyoung²^ORCID,Miller Steven J.³,Nigam Arjun⁴,Onghai Kyle⁵,Saikia Dishant⁶,Sharma Lalit M.⁷

Affiliation:

1. Department of Mathematics, Princeton University, USA

2. Department of Mathematics and Statistics, Queen’s University, Canada

3. Department of Mathematics and Statistics, Williams College, USA

4. Department of Mathematics, University of Arizona, USA

5. Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza Box 951555, Los Angeles, CA 90095, USA

6. Department of Mathematical Sciences, Tezpur University, Tezpur 784028, India

7. Dyal Singh College, University of Delhi, Lodhi Road, Pragati Vihar, Delhi 110003, India

Abstract

In this paper, we define a [Formula: see text]-Diophantine [Formula: see text]-tuple to be a set of [Formula: see text] positive integers such that the product of any [Formula: see text] distinct positive integers is one less than a perfect square. We study these sets in finite fields [Formula: see text] for odd prime [Formula: see text] and guarantee the existence of a [Formula: see text]-Diophantine [Formula: see text]-tuple provided [Formula: see text] is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in [Formula: see text] as well as an asymptotic formula for the number of [Formula: see text]-Diophantine [Formula: see text]-tuples.

Publisher

World Scientific Pub Co Pte Ltd

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S1793042123500458

Reference21 articles.

1. T. M. Apostol , Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics (Springer, New York, 1998), p. 222.

2. THE EQUATIONS 3x2−2 = y2 AND 8x2−7 = z2

3. Nonexistence of D(4)-quintuples

4. There is no Diophantine D(−1)$D(-1)$‐quadruple

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