Diophantine triples

On some particular regular Diophantine 3-tuples

“…Readers can get many significant and valuable information on number theory in the lecture notes of Goldmakher [11], Kurur and Saptharishi [17], and the books of Mollin [18] and Roberts [23]. Besides, one may refer [3,9,10] for an extensive review of various problems on Diophantine m-tuples.…”

A Certain Type of Regular Diophantine Triples and Their Non-Extendability

“…There are lots of papers on the Pell equations and Diophantine n-tuple in the literature. In [1] Beardon & his coauthor and Dujella et al in [2] gave some special and significant results on some of Diophantine triples such as strong Diophantine triples, regular Diophantine triples etc… Gopalan [3,4] worked on both dio special diophantine triples and pellian equation.…”

“…For the proof of theorems, we use the factorization method in set of integers and integer solutions of Pell (Pellian) equations or Pell like equations as well as quadratic resıdue, quadratc reciprocity theorem, legendre symbol, etc... Now, we give some basic notations useful for proving our theorems as follow: Definition 1.1. [1,13] A Diophantine n-tuple with the property D(s) (it sometimes representatives as with n-tuple) for an integer s is a n-tuple of different positive integers { 1 , … , } such that + is always a square of an integer for every distinct i, j.…”

On Sequences of Diophantine 3-Tuples generated through Euler and Bernoulli Polynomials

“…The problem of constructing the sets with property that product of any two of its distinct elements is one less than a square has a very long history and such sets have been studied by Diophantus. for any arbitrary integer n [1] and also, for any linear polynomials in n. In this context, one may refer [2][3][4][5][6][7][8][9][10][11][12] for an extensive review of various problems on diophantine triples. This paper aims at constructing sequences of diophantine triples where the product of any two members of the triple with the polynomial with integer coefficients satisfies the required property.…”