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Adjugates of Diophantine Quadruples
Published/Copyright:
May 31, 2010
Abstract
Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of the set placed in the diagonal and with corresponding roots off-diagonal. In the case of quadruples, Jacobi's theorem for the minors of the adjugate matrix can be used to show that up to eight new Diophantine quadruples can be formed from the adjugate matrices with various combinations of signs for the roots. We call these adjugate quadruples.
Keywords.: Diophantine equation; adjugates; Diophantine quadruples; hyperdeterminant; Jacobi theorem
Received: 2009-07-26
Revised: 2009-12-26
Accepted: 2009-12-30
Published Online: 2010-05-31
Published in Print: 2010-May
© de Gruyter 2010
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- Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions
- Multiplicities of Integer Arrays
- Adjugates of Diophantine Quadruples
- A Binary Tree Representation for the 2-Adic Valuation of a Sequence Arising From a Rational Integral
- On a Variant of Van Der Waerden's Theorem
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- Sequences of Density ζ(k) – 1
- Square-Full Divisors of Square-Full Integers
- Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms
Keywords for this article
Diophantine equation;
adjugates;
Diophantine quadruples;
hyperdeterminant;
Jacobi theorem
Articles in the same Issue
- Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions
- Multiplicities of Integer Arrays
- Adjugates of Diophantine Quadruples
- A Binary Tree Representation for the 2-Adic Valuation of a Sequence Arising From a Rational Integral
- On a Variant of Van Der Waerden's Theorem
- Note on a Result of Haddad and Helou
- Sequences of Density ζ(k) – 1
- Square-Full Divisors of Square-Full Integers
- Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms
