Quadratic residues and \(x^ 3+y^ 3=z^ 3\) in models of \(IE_ 1\) and \(IE_ 2\)
It is unknown whether the quadratic reciprocity law is provable in \(I\Delta_ 0\). The author provides a careful analysis of some elementary number theoretic arguments to show that a certain version of the law can be proven in existential bounded induction, \(IE_ 1\). The result is: if \(M\models IE_ 1\), and \(p\), \(q\) are primes of \(M\), where \(p\) is standard, then \(\bigl({p\over q}\bigr)=1\) implies \(\bigl({q\over p}\bigr)=1\), unless both \(p\) and \(q\) are congruent to 3 modulo 4 in which case \(\bigl({q\over p}\bigr)=-1\). In the other sections of the paper the question of representability of primes of a model of \(IE_ 1\) in the form \(x^ 2+ ny^ 2\), for integer \(n\), is considered, and the results are applied to show that the equation \(x^ 3+ y^ 3= z^ 3\) has no nontrivial solutions in models of \(IE_ 2\).
scientific article; zbMATH DE number 537009