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Bjørn Kjos-Hanssen
Bjørn Kjos-Hanssen
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For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

in binary has the answer Yes. Thanks to @AlexeiKulikov for the improved argument.

Use

  • (i) Prime number theorem
  • (i ii) Pigeonhole principle

By (i) there are at least $c (2^n/n)$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $[1,n]$.

By (ii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.

For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

has the answer Yes. Thanks to @AlexeiKulikov for the improved argument.

Use

  • (i) Prime number theorem
  • (i ii) Pigeonhole principle

By (i) there are at least $c (2^n/n)$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $[1,n]$.

By (ii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.

For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

in binary has the answer Yes. Thanks to @AlexeiKulikov for the improved argument.

Use

  • (i) Prime number theorem
  • (i ii) Pigeonhole principle

By (i) there are at least $c (2^n/n)$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $[1,n]$.

By (ii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.

deleted 130 characters in body
Source Link
Bjørn Kjos-Hanssen
Bjørn Kjos-Hanssen
  • 24.8k
  • 3
  • 58
  • 114

For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

in binary has the answer Yes. Thanks to @AlexeiKulikov for the improved argument.

Use

  • (i) Chebychev's inequality
  • (ii) Prime number theorem
  • (iiii ii) Pigeonhole principle

By (i) for large $n$, most numbers in $[2^{n-1},2^n)$ have $n/2\pm c_1\sqrt{n}$ digits that are 1.

Then by (ii) there are at least $c_2 n$$c (2^n/n)$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $n/2\pm c_1\sqrt{n}$$[1,n]$.

Since $c_2n>2c_1\sqrt{n}$, byBy (iiiii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.

For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

in binary has the answer Yes.

Use

  • (i) Chebychev's inequality
  • (ii) Prime number theorem
  • (iii) Pigeonhole principle

By (i) for large $n$, most numbers in $[2^{n-1},2^n)$ have $n/2\pm c_1\sqrt{n}$ digits that are 1.

Then by (ii) there are at least $c_2 n$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $n/2\pm c_1\sqrt{n}$.

Since $c_2n>2c_1\sqrt{n}$, by (iii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.

For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

has the answer Yes. Thanks to @AlexeiKulikov for the improved argument.

Use

  • (i) Prime number theorem
  • (i ii) Pigeonhole principle

By (i) there are at least $c (2^n/n)$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $[1,n]$.

By (ii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.

Source Link
Bjørn Kjos-Hanssen
Bjørn Kjos-Hanssen
  • 24.8k
  • 3
  • 58
  • 114

For many variants of this question the answer seems to be not known but at least this question in the comments

More generally, are there infinitely many primes such that at least one non-trivial permutation of their digits preserves the primality? – Sylvain JULIEN

in binary has the answer Yes.

Use

  • (i) Chebychev's inequality
  • (ii) Prime number theorem
  • (iii) Pigeonhole principle

By (i) for large $n$, most numbers in $[2^{n-1},2^n)$ have $n/2\pm c_1\sqrt{n}$ digits that are 1.

Then by (ii) there are at least $c_2 n$ primes in $[2^{n-1},2^n)$ whose number of 1s is in the interval $n/2\pm c_1\sqrt{n}$.

Since $c_2n>2c_1\sqrt{n}$, by (iii) there are two primes in $[2^{n-1},2^n)$ with the same number of 1s.