It's important to specify which ring the polynomials are over. If the complex or real numbers then this conjecture must be false for any polynomial with degree at least two, respectively three. So are you thinking over the integers?

Thank you for your answer I was thinking about integers with the degree 1 or above

The degree one condition would be needed in the integers, as there are many counterexamples amongst the constant polynomials. Will have a think about higher degree.

It seems that the article A Goldbach theorem for polynomials with integral coefficients", Amer. Math. Monthly 72 (1965), 45-46 proves what you need, in fact for all non-constant polynomials.

James Tuite

Thanks

My pleasure - a great question! I wonder which other rings this holds for?

Of course in the general case one must distinguish between prime and irreducible, so potentially there are two different questions for a general polynomial ring R[x]

I have to comment

My initial take was to use two prime numbers a and b and make their sum equal aan integer to a certain power of r where r is an integer I certainly did not mean to complicate the problem so if my accompanying file has flaws I really have to apologize but then again that's a reason for my starting this conversation I begin to learn something myself Great feedback though