Erdos 1953 arithmetical properties polynomials

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remarks_p425: The paper's closing remarks, stated without proof: an l-th power plus an l-th power free integer represents every large integer, f(p) is l-th power free for infinitely many primes p, the (l-1)-th power analogue at primes is conjectured, and squarefreeness of n^4+2 infinitely often is left unproved.

theorem: Erdős's theorem that a primitive integer polynomial of degree l >= 3 with positive leading coefficient, not divisible by the (l-1)-th power of an integral linear polynomial and, when l is a power of 2, not having 2^{l-1} divide every value, takes (l-1)-th power free values at infinitely many positive integers; with the variant for the excluded case.

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P. Erdős: Arithmetical properties of polynomials, J. London Math. Soc. 28 (1953), 416--425; MR 15,104f; Zentralblatt 51,277.

Let f be a polynomial of degree l whose integer coefficients have highest common factor 1 and whose leading coefficient is positive. The Theorem (p. 417) proves that if l >= 3, f is not divisible by the (l-1)-th power of a linear polynomial with integral coefficients, and (when l is a power of 2) some n has f(n) not divisible by 2^{l-1}, then f(n) is (l-1)-th power free for infinitely many positive integers n. Section 2 (pp. 417--418) handles the reducible case: when f = phi^k with phi irreducible and k >= 2 it applies the known l-th power result to phi, which is (l/k)-th power free infinitely often, so f(n) is (l-1)-th power free; otherwise it splits f into coprime factors of degree below l and sieves. Sections 3--9 (pp. 418--425) treat irreducible f by a counting argument over values f(k) with a divisor from a set of squarefree integers built from medium-sized primes, resting on Lemmas 1--3 (p. 420). Erdős says that positive density of such n seems very likely but that he has not been able to prove it (p. 417), and notes that in the excluded case f(n) = 0 mod 2^{l-1} for all n the proof gives infinitely many n with f(n) = 2^{l-1} u_n, u_n odd and (l-1)-th power free. The closing remarks (p. 425) state without proof results on an l-th power plus a power free integer and on power free values f(p) at primes, conjecture the (l-1)-th power analogue at primes, and record that Erdős could not prove that n^4+2 is squarefree for infinitely many n.

Read status: claims checked for the Theorem, the excluded-case statement and the closing remarks, read on the page images of the print; the proof was followed but not checked estimate by estimate.

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Bears on. #978: the Theorem (p. 417) proves that f(n) is (l-1)-th power free for infinitely many n, and it covers every polynomial in the problem's first question (irreducible, degree l > 2 not a power of 2); that question asks for positive density of these n, which the paper calls very likely and leaves unproved (p. 417). The closing remarks (p. 425) say Erdős could not prove that n^4+2 is squarefree for infinitely many n, the problem's third question. Apart from n^4+2, the paper says nothing about the second question, at exponent l-2.

Results.

• Theorem (p. 417): under the conditions of §1 and l >= 3, f(n) is (l-1)-th power free for infinitely many positive n; with the excluded-case variant f(n) = 2^{l-1} u_n, u_n odd and (l-1)-th power free.
• Closing remarks (p. 425): unproved statements on sums with power free integers and on power free values at primes, and the unproved squarefreeness of n^4+2 infinitely often.

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