Sinisa Slijepcevic

math.NT — Number Theory

Regularity criteria and an effective bound for irregular extensions of Euler triples

Contributed by Sinisa Slijepcevic, Andrej Dujella

An Euler triple is a Diophantine triple $\{a,b,c\}$ with $a<b<c=a+b+2\sqrt{ab+1}$. It extends to the regular Diophantine quadruple $\{a,b,c,d_+\}$, and it is conjectured that it has no other extension $d>c$. We prove that an irregular extension forces $c<4\cdot10^{48}$, against $c<10^{98}$ in recent work of He and Togb\'e. The proof squeezes the index $m$ of the extension, which locates a common term of two Pell sequences, between a floor from congruences, $c<8m^4$, valid for every irregular extension, and ceilings from linear forms in logarithms. The same squeeze, with other bounds, gives three regularity criteria: by the size of $b/a$, which confines irregular extensions to $1+\frac1{1538}<b/a<3355^2$; by pair coordinates, through the hypergeometric method; and by coprime divisors of the index combinations, which settles every Euler triple with two elements in a ratio $j:k$ with $jk\le300$. As applications, the three-parameter family of Cipu, Fujita and Mignotte extends only regularly when $\nu A\ge2$ and $K\ge643$, and when $K=1$, $A\in\{2,3\}$ and $\nu\ge2$; the latter contains the Hoggatt--Bergum triples, for which this recovers Dujella's theorem for extensions $d>c$.