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A nice trick by Edmund Landau:

Let us suppose that we need to show the validity of Bertrand's postulate for every $n<4000$ (as in Erdös's famous 1932 paper). According to Erdös, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to ponder the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" by Landau (as Erdös refer to it in the aforementioned paper) is definitely deserving of being declared as one, right?

A nice trick by Edmund Landau:

Let us suppose that we need to show the validity of Bertrand's postulate for every $n<4000$ (as in Erdős's famous 1932 paper). According to Erdős, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to ponder the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" by Landau (as Erdős refers to it in the aforementioned paper) is definitely deserving of being declared as one, right?

I am very fond of the device by Landau by means of which one can establish Bertrand's Postulate for every natural number $n$ less than some given constant $C>0$.

Let us suppose that we need to show the validity of the "postulate" for every $n<4000$ (as in Erdös's famous 1932 paper). According to Erdös, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to consider the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" of Landau (as Erdös refer to it in the aforementioned paper) is definitely deserving of being declared as one, right?

A nice trick by Edmund Landau:

Let us suppose that we need to show the validity of Bertrand's postulate for every $n<4000$ (as in Erdös's famous 1932 paper). According to Erdös, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to ponder the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" by Landau (as Erdös refer to it in the aforementioned paper) is definitely deserving of being declared as one, right?

I am very fond of the device by Landau by means of which one can establish Bertrand's Postulate for every natural number $n$ less than some given constant $C>0$.

Let us suppose that we need to show the validity of the "postulate" for every $n<4000$ (as in Erdös's famous 1932 paper). According to Erdös, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], ... (3999, 7998],$$

and that it suffices to consider the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" of Landau (as Erdös refer to it in the aforementioned paper) is definitely deserving of being declared as one, right?

I am very fond of the device by Landau by means of which one can establish Bertrand's Postulate for every natural number $n$ less than some given constant $C>0$.

Let us suppose that we need to show the validity of the "postulate" for every $n<4000$ (as in Erdös's famous 1932 paper). According to Erdös, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to consider the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" of Landau (as Erdös refer to it in the aforementioned paper) is definitely deserving of being declared as one, right?