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Statement

All on p. 425, in the setting of the Theorem ($f(x)$ of degree $l$, integer coefficients with highest common factor $1$, positive leading coefficient). None of these remarks is proved in the paper.

1. Sums. Citing Estermann (Math. Annalen 105 (1931), 653--662) for every sufficiently large positive integer being a square plus a squarefree integer, the paper says similar methods prove that every sufficiently large positive integer is an $l$-th power plus an $l$-th power free integer, and that its own methods would doubtless give an $l$-th power plus an $(l-1)$-th power free integer.
2. Primes. The paper says one can prove that $f(p)$ is $l$-th power free for infinitely many primes $p$, provided $f(x)$ is not the $l$-th power of a linear polynomial. It calls it reasonable to conjecture that $f(p)$ is $(l-1)$-th power free for infinitely many primes $p$ when $f(x)$ satisfies the conditions of §1, and says the methods of the paper do not seem strong enough to prove this.
3. The quartic (quoted). "I have also not been able to prove, for example, that $n^4+2$ is squarefree for infinitely many $n$."

Proof pointer

None: the remarks give no proofs.

Read depth

Claims checked: the remarks, which the paper calls two and which are split here into three items, were read clause by clause on the page image of p. 425. Nothing here is independently reviewed.

Dependencies

None in the corpus. The paper cites Estermann's 1931 paper for the square-plus-squarefree result.

Source. P. Erdős, Arithmetical properties of polynomials, J. London Math. Soc. 28 (1953), 416--425; the edition read is named on the source card.

Bears on

• Problem 978: remark 3 states that the author could not prove that $n^4+2$ is squarefree for infinitely many $n$, which is the problem's third question; the paper proves nothing about it.