Casas-Alvero conjecture: difficulty and analogous conjecture for integers It is well know that some theorems for polynomials have analogous for integers. **Example.** The Mason–Stothers theorem, see this Wikipedia and the abc conjecture for integers. After I've read the statement for fields of characteristic zero of Casas-Alvero conjecture I wondered next questions. > **Question 1.** Can you provide us some idea why is it difficult to prove or refute such the Casas-Alvero conjecture? I am asking from an informative viewpoint, that if you can provide me some idea why such conjecture is so difficult. **I wish you a good day.** For this next question, I don't know if it have full mathematical sense. If this is the issue please explain why the analogous Casas-Alvero conjecture for positive integers have no mathematical meaning. > **Question 2.** Has mathematical meaning and what should be an analogous statement , than Casas-Alvero conjecture, for integers? **Many thanks.**

_Question 1_ : There is a good survey in the article Constraints on counterexamples to the Casas-Alvero conjecture, and a verification in degree 12 , which gives some idea why it is difficult to prove or refute this conjecture. Phrases like "Because of a lack of a general strategy" support this, too. The conjecture has been proven for prime power degree. The smallest unknown cases are degrees $12, 20, 24$ and $28$.

_Question 2_ : An analogue for integers could start from the arithmetic derivative, which is a version of "derivative" for integers. I have not seen a "reasonable" version of Casas-Alvero for integers so far, in contrast to Fermat for polynomials, Mason-Stothers and Waring's problem.