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Living Hell
Living Hell
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Suppose that someone proves that there exist a sequence (if that is not already known) of the form $a^n+b$ where $a \in \mathbb N$$a \in \mathbb N \setminus \{1\}$ and $b \in \mathbb Z$ which has an infinite number of primes as its values.

Now let us define the set $S_k=\{1 \leq i \leq k: a^i+b\in \mathbb P\}$. Let us denote with $|S_k|$ the cardinality of $S_k$.

Do we have $\lim_{k \to \infty} \dfrac {|S_k|}{k} =0$?

In other words, even if there is no concrete example of the sequence of the form $a^n+b$ which has an infinite number of primes as its values can it be proven that the density of primes in such sequences always equals zero?

Suppose that someone proves that there exist a sequence (if that is not already known) of the form $a^n+b$ where $a \in \mathbb N$ and $b \in \mathbb Z$ which has an infinite number of primes as its values.

Now let us define the set $S_k=\{1 \leq i \leq k: a^i+b\in \mathbb P\}$. Let us denote with $|S_k|$ the cardinality of $S_k$.

Do we have $\lim_{k \to \infty} \dfrac {|S_k|}{k} =0$?

In other words, even if there is no concrete example of the sequence of the form $a^n+b$ which has an infinite number of primes as its values can it be proven that the density of primes in such sequences always equals zero?

Suppose that someone proves that there exist a sequence (if that is not already known) of the form $a^n+b$ where $a \in \mathbb N \setminus \{1\}$ and $b \in \mathbb Z$ which has an infinite number of primes as its values.

Now let us define the set $S_k=\{1 \leq i \leq k: a^i+b\in \mathbb P\}$. Let us denote with $|S_k|$ the cardinality of $S_k$.

Do we have $\lim_{k \to \infty} \dfrac {|S_k|}{k} =0$?

In other words, even if there is no concrete example of the sequence of the form $a^n+b$ which has an infinite number of primes as its values can it be proven that the density of primes in such sequences always equals zero?

Source Link
Living Hell
Living Hell
  • 221
  • 1
  • 6

Density of primes in sequences of the form $a^n+b$

Suppose that someone proves that there exist a sequence (if that is not already known) of the form $a^n+b$ where $a \in \mathbb N$ and $b \in \mathbb Z$ which has an infinite number of primes as its values.

Now let us define the set $S_k=\{1 \leq i \leq k: a^i+b\in \mathbb P\}$. Let us denote with $|S_k|$ the cardinality of $S_k$.

Do we have $\lim_{k \to \infty} \dfrac {|S_k|}{k} =0$?

In other words, even if there is no concrete example of the sequence of the form $a^n+b$ which has an infinite number of primes as its values can it be proven that the density of primes in such sequences always equals zero?