engleski

Extensions of a Diophantine triple by adjoining smaller elements

In this paper, we prove that if {; ; ; a1, b, c, d}; ; ; and {; ; ; a2, b, c, d}; ; ; are Diophantine quadruples with a1 < a2 < b < c < d, then a2 > 24^3 , a2 > max{; ; ; 36a1^3 , 300a1^2}; ; ; , b < a2^{; ; ; 3/2}; ; ; , and 16a1^2 b^3 < c < 16a2 b^3. The last inequalities imply that for a fixed Diophantine triple {; ; ; b, c, d}; ; ; the number of Diophantine quadruples {; ; ; a, b, c, d}; ; ; with a < min{; ; ; b, c, d}; ; ; is at most two. Moreover, we show that there are only finitely many quintuples {; ; ; a1, a2, b, c, d}; ; ; as above.

Diophantine m-tuples, Pellian equations, hypergeometric method

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