On Diophantine quintuples and $$D(-1)$$ D ( - 1 ) -quadruples

Explicit upper bound for the average number of divisors of irreducible quadratic polynomials

“…It is conjectured that there are no -quadruples but currently it is only known that there are at most -quadruples [11]. The latter number upgraded the previous maximal possible bound due to Trudgian [19], which in turn improved results from [2, 6]. …”

“…Plugging the upper bound of Corollary 3 in the proof of ([6], Theorem 1.3) from the paper of Elsholtz, Filipin and Fujita we obtain another slight improvement.…”

On the average number of divisors of reducible quadratic polynomials

“…Previous explicit upper bounds for the average number of divisors of P (n) = n 2 − 1 were obtained by Elsholtz, Filipin and Fujita [6] with A(1) ≤ 2. Trudgian [21] improved this to A(1) ≤ 12/π 2 , Cipu [3] got A(1) ≤ 9/π 2 and very recently Cipu and Trudgian [4] also achieved the best leading coefficient A(1) ≤ 6/π 2 using different method than ours.…”

Bounds for Diophantine quintuples

“…In [8] Dujella obtained results very close to settling this conjecture by proving that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. Later bounds on the number of Diophantine quintuples are provided in [9,11,16]. Historical and recent developments of the study of Diophantine m-tuples are found on Dujella's webpage: http://web.math.pmf.unizg.hr/∼duje/dtuples.html.…”