A weak effective \(abc\)-conjecture
For a triple \((a,b,c)\) of relatively prime positive integers satisfying \(a+b=c\) define \[ L=L(a,b,c)={\log c \over \log \text{rad}(a,b,c)}, \] where the radical \(\text{rad} (m)\) of a positive integer is the product of the distinct prime factors of \(m\). The \(abc\)--conjecture states that \(\limsup L(a,b,c)=1\), where \((a,b,c)\) runs over all triples as above. The \textit{weak effective \(abc(r)\)-conjecture} for a given real \(r\leq 1.5\) says that all triples \((a,b,c)\) satisfying \(L(a,b,c)>r\) are the presently calculated triples (given in a table of the paper). Applications are given for several classical Diophantine equations, for example for the Nagell-Ljunggren equation \((x^n-1)/(x-1)=y^q\) the author shows that the above conjecture for \(r=145/63)=2.3015\ldots\) implies that this equation has only the three presently known solutions, \((x,n,y,q)=(3,5,11,2)=(7,4,20,2)=(18,3,7,3)\).
scientific article; zbMATH DE number 5635540