Can any large number $N\in \mathbb{Z}$ be written as a sum of a square and a square-free?

Question

I'm reading an article by Dan Carmon on square free values of large polynomials over the rational function field. In this article he states the following question:

Does every sufficiently large $N\in \mathbb{Z}$ admit a representation as a sum $N = x^k+r$ of a positive $k$-th power and a positive square-free? How many such representations are there, asymptotically?

He then states that this has been proven for $k=2$ and $k=3$ by Estermann. I'm trying to find these proofs, but I'm not yet succeeding. Any idea how to prove this or where I can find the proofs?

Answer 1

Take a look at this short paper of Erdös:

The representation of an integer as the sum of the square of a prime and of a square-free integer (free-pdf), J. London Math. Soc. 10 (1935), 243-245 .

At the end of the paper he states that "We can prove similarly the more general theorem $n=p^k+g$, where $k$ is a given exponent and $g$ is free from $k$-th power divisors."

P.S. T. Estermann, "Einige Sätze über quadratfreie Zahlen", Math. Annalen, 105 (1931), 653-662.