Problems & Puzzles: Puzzles

Puzzle 1127 Prime(11!)+Prime(11)!   On March 24, 2023 Sebastián Martín Ruiz wrote: I have found this curio prime: Prime(11!)+Prime(11)! is prime. Q. Are there more primes of this form?

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During the week from 2-8, April 2023, contributions came from Michael Branicky, Oscar Volpatti

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Michael wrote:

I found no other primes for n = 1..21.

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Oscar wrote:

I checked some more sums up to  Prime(21!)+Prime(21)! but I found no more primes.

Computing Prime(n!) gets harder and harder; for n>15, I used values from OEIS sequence A062439.

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On April, 14, 2023, Oscar Volpatti wrote:

I computed next term a(22) = Prime(22!) of OEIS sequence A062439:
Prime(22!) = 57762800476687420222169.

No new primes for puzzles 1127 and 1128:
Prime(22)! + Prime(22!) is composite,
Prime(22)! - Prime(22!) is composite too.

I used the program "primecount" by Kim Walisch.
Quoting his explanation about fast nth prime computation.
"The most efficient known method for calculating the nth prime is a combination of the prime counting function and a prime sieve.
The idea is to closely approximate the nth prime e.g. using the inverse logarithmic integral or the inverse Riemann R function and then count the primes up to this guess using the prime counting function.
Once this is done one starts sieving (e.g. using the segmented sieve of Eratosthenes) from there on until one finds the actual nth prime."

Target index:
n = 22! = 1124000727777607680000.
Guess 1, using the inverse logarithmic integral:
x_1 = 57762800476481495425024.
Guess 2, using the inverse Riemann R function:
x_2 = 57762800476731842462240.
Actual number of primes up to guess 2:
n_2 = Pi(x_2) = 1124000727778455251835.
Sieving backwards:
x = 57762800476687420222169 is prime,
Pi(x) = n,
Pi(x-1) = n-1,

nth prime found.

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