Euler Problem 7: 10,001st Prime and Sexy Primes

Euler problem 7 asks us to generate ten thousand prime numbers.

Euler Problem 7 Definition

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. What is the 1,0001st prime number?

Solution

TheĀ is.prime function determines whether a number is a prime number by checking that it is not divisible by any prime number up to the square root of the number. The Sieve of used in Euler Problem 3 generates the prime numbers. This problem can only be solved using brute force becauseĀ prime gaps (sequence A001223 in the OEIS) do not follow a predictable pattern. You can download the latest version of this code from GitHub.

is.prime <- function(n) {
    primes <- esieve(ceiling(sqrt(n)))
    prod(n %% primes! = 0) == 1
}
i <- 2 # First Prime
n <- 1 # Start counter
while (n < 10001) { # Find 10001 prime numbers
    i <- i + 1 # Next number
    if(is.prime(i)) { # Test next number
        n <- n + 1 # Increment counter
        i <- i + 1 # Next prime is at least two away
    }
}
answer <- i-1

Sexy Primes

The largest prime gap for the first 10,001 primes is 72. Sexy primes with a gap of 6 are the most common and there are 1270 twin primes.

Euler Problem 7: 10,001st Prime and Sexy Primes
The frequency of prime gaps for the first 10,000 primes.
library(tidyverse)
primes <- esieve(i)
p <- length(primes)
gaps <- data_frame(Gap = primes[2:p] - primes[1:(p - 1)], Sexy = Gap %% 6 == 0) %>%
    group_by(Gap, Sexy) %>%
    count()

ggplot(gaps, aes(Gap, n, fill = Sexy)) +
    geom_col() +
    scale_fill_manual(values = c( "#7391AB", "#A62102")) +
    labs(title = "Frequency of prime gaps for the first 10,000 primes",
         x = "Prime Gap",
         y = "Frequency") 
guide_legend(title = "Sexy Prime")

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