The number of irregular Diophantine quadruples for a fixed Diophantine pair or triple. (English) Zbl 1520.11040
Adamović, Dražen (ed.) et al., Lie groups, number theory, and vertex algebras. Representation theory XVI. Conference, Inter-University Center, Dubrovnik, Croatia, June 24–29, 2019. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 768, 105-117 (2021).
A set of \( m \) positive integers \( \{a_1, \ldots, a_m\} \) is called a Diophantine \( m \)-triple if \( a_ia_j+1 \) is a perfect square for all \( i,j \) with \( 1\le i<j\le m \). A special case of the Diophantine quadruple \( \{a,b,c,d_+\} \) for a Diophantine triple \( \{a,b,c\} \) is called regular if \(d_+:=d_+(a,b,c)=a+b+c+2abc+2rst\), where \((r,s,t)=\left(\sqrt{ab+1}, \sqrt{ac+1}, \sqrt{bc+1}\right)\). It can also be noted that if \( d_-:=d_-(a,b,c) =a+b+c+2abc-2rst\) for a Diophantine triple \( \{a,b,c\} \), then any of \( ad_-+1 \), \( bd_-+1 \) and \( cd_-+1 \) is a perfect square. Thus, if \( c>a+b+2r \), then \( \{a,b,c,d_-\} \) is a regular Diophantine quadruple with \( c=d_+(a,b,d_-) \). Any Diophantine quadruple that is not regular is classified as irregular. The main aim of the paper under review is to give upper bounds for the number of irregular Diophantine quadruples \( \{a,b,c,d\} \) for a fixed Diophantine pair \( \{a,b\} \) or Diophantine triple \( \{a,b,c\} \) without assuming that \( \max\{a,b,c\}<d \).
For the entire collection see [Zbl 1470.17001].
For the entire collection see [Zbl 1470.17001].
Reviewer: Mahadi Ddamulira (Kampala)
MSC:
| 11D45 | Counting solutions of Diophantine equations |
| 11D09 | Quadratic and bilinear Diophantine equations |
| 11B37 | Recurrences |
Software:
PARI/GPReferences:
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