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Vesselin Dimitrov
Vesselin Dimitrov
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You want an elementary argument for the existence of an infinite set of primes $p$ which at once divide a number of the form $x^4-2$ and a number of the form $y^4+1$, where $x,y \in \mathbb{Z}$. Here is a recipe which expands on Timo Keller's hint.

I claim that there are rational polynomials $U,V \in \mathbb{Q}[X]$ such that $U^4-2$ and $V^4+1$ share a common non-constant factor $P \in \mathbb{Q}[X]$. To see the claim, just consider an $\alpha \in \bar{\mathbb{Q}}$$\alpha \in \mathbb{Q}(\sqrt[4]{2},e^{\pi i/4})$ such that $\mathbb{Q}(\sqrt[4]{2},e^{\pi i/4}) = \mathbb{Q}(\alpha)$,; for example, $\alpha := \sqrt[4]{2}+e^{\pi i/4}$ will do. Then you can express $\sqrt[4]{2} = U(\alpha)$ and $e^{\pi i/4} = V(\alpha)$ with $U,V \in \mathbb{Q}[X]$. Now both polynomials $U^4-2$ and $V^4+1$ are divisible in $\mathbb{Q}[X]$ by the minimum polynomial $P(X) \in \mathbb{Q}[X]$ of $\alpha$.

It is thus sufficient to consider (large enough) primes which divide some value $P(n)$, $n \in \mathbb{Z}$. You know how to conclude a la Euclid: if $p_1,\ldots,p_k$ were all such primes, the consideration of $P\big(Np_1\cdots p_k\big)$ would lead you to contradiction for $N \gg 0$.

You want an elementary argument for the existence of an infinite set of primes $p$ which at once divide a number of the form $x^4-2$ and a number of the form $y^4+1$, where $x,y \in \mathbb{Z}$. Here is a recipe which expands on Timo Keller's hint.

I claim that there are rational polynomials $U,V \in \mathbb{Q}[X]$ such that $U^4-2$ and $V^4+1$ share a common non-constant factor $P \in \mathbb{Q}[X]$. To see the claim, just consider an $\alpha \in \bar{\mathbb{Q}}$ such that $\mathbb{Q}(\sqrt[4]{2},e^{\pi i/4}) = \mathbb{Q}(\alpha)$, for example $\alpha := \sqrt[4]{2}+e^{\pi i/4}$. Then you can express $\sqrt[4]{2} = U(\alpha)$ and $e^{\pi i/4} = V(\alpha)$ with $U,V \in \mathbb{Q}[X]$. Now both polynomials $U^4-2$ and $V^4+1$ are divisible in $\mathbb{Q}[X]$ by the minimum polynomial $P(X) \in \mathbb{Q}[X]$ of $\alpha$.

It is thus sufficient to consider (large enough) primes which divide some value $P(n)$, $n \in \mathbb{Z}$. You know how to conclude a la Euclid: if $p_1,\ldots,p_k$ were all such primes, the consideration of $P\big(Np_1\cdots p_k\big)$ would lead you to contradiction for $N \gg 0$.

You want an elementary argument for the existence of an infinite set of primes $p$ which at once divide a number of the form $x^4-2$ and a number of the form $y^4+1$, where $x,y \in \mathbb{Z}$. Here is a recipe which expands on Timo Keller's hint.

I claim that there are rational polynomials $U,V \in \mathbb{Q}[X]$ such that $U^4-2$ and $V^4+1$ share a common non-constant factor $P \in \mathbb{Q}[X]$. To see the claim, just consider an $\alpha \in \mathbb{Q}(\sqrt[4]{2},e^{\pi i/4})$ such that $\mathbb{Q}(\sqrt[4]{2},e^{\pi i/4}) = \mathbb{Q}(\alpha)$; for example, $\alpha := \sqrt[4]{2}+e^{\pi i/4}$ will do. Then you can express $\sqrt[4]{2} = U(\alpha)$ and $e^{\pi i/4} = V(\alpha)$ with $U,V \in \mathbb{Q}[X]$. Now both polynomials $U^4-2$ and $V^4+1$ are divisible in $\mathbb{Q}[X]$ by the minimum polynomial $P(X) \in \mathbb{Q}[X]$ of $\alpha$.

It is thus sufficient to consider (large enough) primes which divide some value $P(n)$, $n \in \mathbb{Z}$. You know how to conclude a la Euclid: if $p_1,\ldots,p_k$ were all such primes, the consideration of $P\big(Np_1\cdots p_k\big)$ would lead you to contradiction for $N \gg 0$.

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Vesselin Dimitrov
Vesselin Dimitrov
  • 13.8k
  • 3
  • 56
  • 95

You want an elementary argument for the existence of an infinite set of primes $p$ which at once divide a number of the form $x^4-2$ and a number of the form $y^4+1$, where $x,y \in \mathbb{Z}$. Here is a recipe which expands on Timo Keller's hint.

I claim that there are rational polynomials $U,V \in \mathbb{Q}[X]$ such that $U^4-2$ and $V^4+1$ share a common non-constant factor $P \in \mathbb{Q}[X]$. To see the claim, just consider an $\alpha \in \bar{\mathbb{Q}}$ such that $\mathbb{Q}(\sqrt[4]{2},e^{\pi i/4}) = \mathbb{Q}(\alpha)$, for example $\alpha := \sqrt[4]{2}+e^{\pi i/4}$. Then you can express $\sqrt[4]{2} = U(\alpha)$ and $e^{\pi i/4} = V(\alpha)$ with $U,V \in \mathbb{Q}[X]$. Now both polynomials $U^4-2$ and $V^4+1$ are divisible in $\mathbb{Q}[X]$ by the minimum polynomial $P(X) \in \mathbb{Q}[X]$ of $\alpha$.

It is thus sufficient to consider (large enough) primes which divide some value $P(n)$, $n \in \mathbb{Z}$. You know how to conclude a la Euclid: if $p_1,\ldots,p_k$ were all such primes, the consideration of $P\big(Np_1\cdots p_k\big)$ would lead you to contradiction for $N \gg 0$.