The Real Part of Mihăilescu’s Ideal

In this chapter we continue our study of Mihăilescu’s ideal. As follows from the definition, it contains the ideal q ℤ [ G ] $$q\mathbb{Z}[G]$$ of the elements divisible by q. A basic question is whether Mihăilescu’s ideal has nontrivial (that is, not divisible by q) elements.In this chapter we prove that (for large x) the real part ℐ M + $$\mathcal{I}_{M}^{+}$$ of Mihăilescu’s ideal contains no nontrivial elements of weight 0 and even of any weight divisible by q.On the other hand, later we shall see that a solution to Catalan’s equation gives rise to a nontrivial element of ℐ M + $$\mathcal{I}_{M}^{+}$$ of weight divisible by q. This contradiction would prove Catalan’s conjecture.