Problems & Puzzles: Puzzles

Puzzle 1006. Bemirps as index of the π(i), the prime counting function.

G. L. Honaker, Jr. made me notice the following curio by Sariyar:

"The smallest bemirp such that 3 of the 4 associated primes are prime-indexed primes: π(1888691180161) = 69368137291 is prime, π(1888961180191) = 69377688539 is prime, π(1910811698881) = 70150532951 is prime. Unfortunately, π(161081196881) = 59510529369 = 3^2 883 7488427. [Sariyar]"

A bemirp (or bi-directional emirp) is a prime that yields a different prime when turned upside down with reversals of each being two more different primes. The result is four different associated primes. E.g., 1061 turned upside down yields 1901, and the reversals of these are 1601 and 1091 respectively.

 

Sariyar discovered that 1888691180161 is a bemirp because

the following four are primes:
 

1888691180161

1888961180191

1910811698881

1610811968881
 

Morover, Sariyar discovered the π(x) is prime when x is sustituted by the first three values:

 

π(1888691180161) = 69368137291 is prime
π(1888961180191) = 69377688539 is prime
π(1910811698881) = 70150532951 is prime.

 

But

 

π(161081196881) = 59510529369 is a composite equal to
3^2 883 7488427 
 

Q. Can you find a bemirp whose four π counting functions are primes?

 


During the week from 20-26 June, 2020, contributions came from Giovanni Resta

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Giovanni wrote:

The smallest set of such primes should be:

pi(10106099866009181) = 282117839673749
pi(10109066899006181) = 282198351694327
pi(18160099866090101) = 498782426824111
pi(18190066899060101) = 499582853840743

where the indices 282117839673749, 282198351694327, 498782426824111,
499582853840743 are 4 prime numbers.

and another set (it may not be the next ) is :

pi(16661919111901891) = 458719357590973
pi(19991616111601861) = 547640065672103
pi(16810611161619991) = 462699726152077
pi(19810911191916661) = 542825062967851

These have been found with the help of Primecount,
a program by Kim Walisch : https://github.com/kimwalisch/primecount
that can compute efficiently pi(n) and also prime(n).
 

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