We study two representation problems in number theory:

(I) Erd?pan>s and Sur᮹i proved that every integer n can be represented in infinitely many ways in the form for some positive integer m and some choice of signs + or ?. We extend theirresult for representations by cubes and fourth powers. To this end, for each of these two powers, we reduce the problem to that of verifying a finite number of cases, which we solve using a computer search program we tailored for this task.

(II) By a result of Hayes, every polynomial in is a sum of two irreducible polynomials over We present an efficient algorithm which, given a polynomial with integer coefficients and a pair of distinct primes, computes a decomposition of as a sum of two irreducible polynomials (of the same degree). For a suitable infinite set of pairs of primes, all such decompositions can be guaranteed to be different, and thereby we obtain the result that every polynomial can be written as a sum of two irreducible polynomials in infinitely many ways.

additive number theory polynomials, irreducibility, infinitely many representations.