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Konstantinos Gaitanas
Konstantinos Gaitanas
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I would like to ask the following question:

Is it possible to find two non-constant polynomials $p(x), q(x)$ with integer coefficients, such that $\gcd(p(n), q(m))=1$ for every $(n, m)\in \mathbb{N}^2$?

If such $p(x), q(x)$ exist, we will call them "completely" coprime, since all of their values will be coprime.
Obviously $p(x), q(x)$ must not have the same root, but this does not seem to help. The problem seems to be quite simple and I suspect that the answer is no, but I was unable to prove this.

I would like to ask the following question:

Is it possible to find two polynomials $p(x), q(x)$ with integer coefficients, such that $\gcd(p(n), q(m))=1$ for every $(n, m)\in \mathbb{N}^2$?

If such $p(x), q(x)$ exist, we will call them "completely" coprime, since all of their values will be coprime.
Obviously $p(x), q(x)$ must not have the same root, but this does not seem to help. The problem seems to be quite simple and I suspect that the answer is no, but I was unable to prove this.

I would like to ask the following question:

Is it possible to find two non-constant polynomials $p(x), q(x)$ with integer coefficients, such that $\gcd(p(n), q(m))=1$ for every $(n, m)\in \mathbb{N}^2$?

If such $p(x), q(x)$ exist, we will call them "completely" coprime, since all of their values will be coprime.
Obviously $p(x), q(x)$ must not have the same root, but this does not seem to help. The problem seems to be quite simple and I suspect that the answer is no, but I was unable to prove this.

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Konstantinos Gaitanas
Konstantinos Gaitanas
  • 3.9k
  • 1
  • 24
  • 39

Two polynomials which are "completely" coprime

I would like to ask the following question:

Is it possible to find two polynomials $p(x), q(x)$ with integer coefficients, such that $\gcd(p(n), q(m))=1$ for every $(n, m)\in \mathbb{N}^2$?

If such $p(x), q(x)$ exist, we will call them "completely" coprime, since all of their values will be coprime.
Obviously $p(x), q(x)$ must not have the same root, but this does not seem to help. The problem seems to be quite simple and I suspect that the answer is no, but I was unable to prove this.