An irregular D(4)-quadruple cannot be extended to a quintuple

Two-parameter families of uniquely extendable Diophantine triples

“…A close similarity of results on D(1)-and D(4)-sets is well documented in literature, as found, e.g., by comparing [7] and [19] with [11,14] and [12]. One of the common properties is that any D(σ)-triple with σ ∈ {1, 4} can be extended to a D(σ)-quadruple.…”

“…Since for the particular triple we are studying one has cr − st = 4, it remains to show that one cannot have |z 0 | < 1.608a −5/14 c 9/14 . Assuming the contrary, it results {a, (z 2 0 − 4)/c, b, c} is a D(4)-quadruple to which Proposition 1 in [12] applies, giving c > min{0.173b 13/2 a 11/2 , 0.087b 7/2 a 5/2 }, which is obviously false.…”

“…[1,9]). Among d's such that {a, b, c, d} is a D(σ)quadruple with a < b < c < d, the smallest integer is known to be d + from [7,Proposition 1] and [12,Proposition 1].…”

On the average number of divisors of reducible quadratic polynomials

“…We believe that in a similar way the upper bounds which will be given by Theorem 3 and Corollary 4 stated below can be useful for estimating the number of D(4), D (16) or other D(k 2 ) -sets, which are investigated in a number of papers, e.g. [1], [9], [10], [11], [12].…”

On a family of two-parametric D(4)-triples

“…The first author studied the size of a D(4)-m-tuple. He proved that there does not exist a D(4)-sextuple and that there are only finitely D(4)-quintuples (see [8][9][10][11]). …”