On a family of Diophantine triples \(\{k,A^2k+2A,(A+1)^2k+2(A+1)\}\) with two parameters

DOI10.1007/s10474-009-8155-5zbMath1199.11074OpenAlexW2137672922MaRDI QIDQ1046923

Bo He, Alain S. Togbé

Publication date: 29 December 2009

Published in: Acta Mathematica Hungarica (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10474-009-8155-5

zbMATH Keywords

Pell equationBaker's methodDiophantine \(m\)-tuple

Mathematics Subject Classification ID

Quadratic and bilinear Diophantine equations (11D09) Cubic and quartic Diophantine equations (11D25) Linear forms in logarithms; Baker's method (11J86)

Related Items

The extensibility of the \(D(\pm k)\)-triple \(\{k\mp 1,k, 4k\mp 1\}\), The number of irregular Diophantine quadruples for a fixed Diophantine pair or triple, The extension of the \(D(-k^2)\)-pair \(\left\{k^2, k^2+1\right\}\), On the \(D(- 1)\)-triple \(\{ 1,k^{2}+1,k^{2}+2k+2\}\) and its unique \(D(1)\)-extension, On a family of Diophantine triples \(\{K,A^2K + 2A, (A + 1)^2 K + 2(A + 1)\}\) with two parameters. II, The regularity of Diophantine quadruples, There is no Diophantine quintuple, Two-parameter families of uniquely extendable Diophantine triples, Extensions of the \(D(\mp k^2)\)-triples \(\{k^2,k^2 \pm 1, 4k^2 \pm 1\}\), Diophantine triples with largest two elements in common, Bounds on the number of Diophantine quintuples, The extendibility of Diophantine pairs. II: Examples

Cites Work

• There does not exist a \(D(4)\)-sextuple
• Any Diophantine quintuple contains a regular Diophantine quadruple
• The Hoggatt-Bergum conjecture on \(D(-1)\)-triples \(\{F_{2k+1}\), \(F_{2k+3}\), \(F_{2k+5}\}\) and integer points on the attached elliptic curves
• An absolute bound for the size of Diophantine \(m\)-tuples
• Diophantine \(m\)-tuples and elliptic curves
• On the size of sets in which \(xy + 4\) is always a square
• The extensibility of Diophantine pairs \(\{k - 1,k+1\}\)
• Nonextendibility of \(D(-1)\)-triples of the form \(\{1,10,c\}\)
• An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II
• On the size of Diophantine m-tuples
• The number of Diophantine quintuples
• A VARIATION ON A PROBLEM OF DAVENPORT AND DIOPHANTUS
• A proof of the Hoggatt-Bergum conjecture
• Simultaneous rational approximations and related diophantine equations
• Generalization of a problem of Diophantus
• There are only finitely many Diophantine quintuples
• COMPLETE SOLUTION OF A PROBLEM OF DIOPHANTUS AND EULER
• The non-extensibility of D(4k)-triples {1, 4k(k-1), 4k^2+1} with |k| prime
• Effective solution of the D(-1)-quadruple conjecture
• THE EQUATIONS 3x²−2 = y² AND 8x²−7 = z²
• ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k³− 4k}
• On the number of solutions of simultaneous Pell equations
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item