The Billaud Conjecture, first stated in 1993, is a fundamental problem on finite words and their heirs, i.e., the words obtained by a projection deleting a single letter. The conjecture states that every morphically primitive word, i.e., a word that is not a fixed point of any non-identity morphism, has at least one morphically primitive heir. The correctness of the conjecture has so far been established in a few special cases, which mainly restrict the alphabet size. In this paper we give a proof for the next such case, i.e., for alphabet size 4.

The Billaud Conjecture, which has been open since 1993, is a fundamental problem on finite words w and their heirs, i.e., the words obtained by deleting every occurrence of a given letter from w. It posits that every morphically primitive word, i.e., a word which is a fixed point of the identity morphism only, has at least one morphically primitive heir. In this paper, we introduce and investigate the related class of so-called Billaud words, i.e., words whose all heirs are morphically imprimitive. We provide a characterisation of morphically imprimitive Billaud words, using a new concept. We show that there are two phenomena through which words can have morphically imprimitive heirs, and we highlight that only one of those occurs in morphically primitive words. Finally, we examine our concept further, and we use it to rephrase and study the Billaud Conjecture in more detail.