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The claim about f of degree 4 or greater should have been for irreducible f.
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Bjorn Poonen
Bjorn Poonen
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More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of C. Hooley). In particular, this answers your question for $f(x)=x^2+k$. There is no irreducible $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, A. Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21-26.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.

More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of C. Hooley). In particular, this answers your question for $f(x)=x^2+k$. There is no $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, A. Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21-26.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.

More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of C. Hooley). In particular, this answers your question for $f(x)=x^2+k$. There is no irreducible $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, A. Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21-26.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.

Erdos --> Hooley, added ref and link
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Bjorn Poonen
Bjorn Poonen
  • 23.8k
  • 7
  • 90
  • 109

More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of ErdosC. Hooley). In particular, this answers your question for $f(x)=x^2+k$. There is no $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, AndrewA. Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21-26.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomialsSquarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.

More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of Erdos). In particular, this answers your question for $f(x)=x^2+k$. There is no $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, Andrew Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.

More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of C. Hooley). In particular, this answers your question for $f(x)=x^2+k$. There is no $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, A. Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21-26.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.

Source Link
Bjorn Poonen
Bjorn Poonen
  • 23.8k
  • 7
  • 90
  • 109

More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c_p p^{-2}$, and these conditions should be independent by the Chinese remainder theorem, so one would conjecture that the fraction of integers $x$ in $\{1,2,\ldots,N\}$ such that $f(x)$ is squarefree should tend to $\prod_p (1-c_p p^{-2})$, where the product is taken over all primes $p$. For large $p$, we have $c_p \le \deg f$, so this product converges.

This guess has been proved for $\deg f \le 3$ (the case $\deg f=3$ is a nontrivial result of Erdos). In particular, this answers your question for $f(x)=x^2+k$. There is no $f$ of degree $4$ or greater for which the density is known to exist (except in cases when the density is $0$ because some $c_p$ equals $p^2$). On the other hand, Andrew Granville proved that the $abc$ conjecture implies that the density exists and equals the predicted value for $f$ of any degree. For further references and a generalization to multivariable polynomials, see the papers cited in the references below.

A. Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 1998, no. 19, 991-1009.

B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353-373.