Number Theory

Adjugates of Diophantine Quadruples

Authors: Philip Gibbs
Comments: 7 pages. Published in INTEGERS 10 (2010), 201-209, (The Electronic Journal of Combinatorial Number Theory)

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.
Category: Number Theory