Step by Step solution about How to resolve the algorithm Long primes step by step in the Julia programming language

The provided Julia code is designed to find and work with "long primes," a specific type of prime numbers that exhibit a particular property. A step-by-step explanation of the code is given below:

1. The code begins by importing the Primes module, which provides functionality for working with prime numbers in Julia.

2. divisors(n) function:

   • This function takes a positive integer n as input.
   • It calculates all the divisors of n and returns them as an array.

3. islongprime(p) function:

   • This function takes a prime number p as input and checks if it is a "long prime."
   • It first finds all the divisors of p-1.
   • Then, for each divisor i, it checks if 10^i % p == 1. If this condition holds for any divisor i, the function returns true, indicating that p is a long prime.
   • Otherwise, it returns false.

4. The code then iterates over all prime numbers less than or equal to 500 and checks if each prime is a long prime using the islongprime function.

5. If a prime p is found to be a long prime, it is printed to the console.

6. Finally, the code calculates and prints the number of long primes for several larger limits, such as 1000, 2000, 4000, and so on, up to 64000.

In summary, this code identifies and counts "long primes" for different limits, where a long prime is a prime number whose previous number (p-1) has a divisor i such that 10^i % p == 1.

using Primes

function divisors(n)
    f = [one(n)]
    for (p,e) in factor(n)
        f = reduce(vcat, [f*p^j for j in 1:e], init=f)
    end
    return length(f) == 1 ? [one(n), n] : sort!(f)
end
 
function islongprime(p)
    for i in divisors(p-1)
        if powermod(10, i, p) == 1
            return i + 1 == p
        end
    end
    false
end
 
println("Long primes ≤ 500: ")
for i in 2:500
    if islongprime(i)
        i == 229 ? println(i) : print(i, "  ")
    end
end
print("\n\n")
 
for i in [500, 1000, 2000, 4000, 8000, 16000, 32000, 64000]
    println("Number of long primes ≤ $i: $(sum(map(x->islongprime(x), 1:i)))")
end